Classic racing games.

GREAT RACING GAMES  (Circuit racing, not dodge-the-traffic kind of racing):

Your experience of a game is greatly influenced by which peripheral you use. Most of these games were played with a d-pad. I haven't yet found an analog-stick controller that is accurate and responsive enough.

RalliSport Challenge 2 + DLC: arcade style with depth in track mastering. Has LAN.
Micro Machines 2TT (Genesis): 4-player action with miniature tire squealers. Rock & Ro!!
Gran Turismo 2 v1.2: today's racing games can learn a lot from GT2, including all GT sequels.
F1 '05 + F1 CE: F1 '05 is more fun than F1 '06, but F1 '06 is more realistic. F1 CE for LAN.
MotoGP 19 Retail: The retail release version is absolutely awesome. Has no LAN.
WRC 3 + 4 + Rally Evolved: they became more realistic until Evolved and are still worth playing.
Gran Turismo 4: good LAN + steering wheel game.
Project Gotham Racing 4: excellent bike game, great multi-player. Has LAN.
Dirt Rally: not an easy game, but rally lovers will come back to it time and again. DR2 for WRX.
Grid Autosport CE: use the PC version with camera-mod, skins & linearity -10 (has no LAN).
Toca 2: (physics of PC version is too simplified) good single and multi-player game (link cable).
ForzaM 4: best FM, but huge braking distances (avoid Horizons). Use DTM Opel. FM2 for LAN.
RIDE 3 Retail: It is similar to MotoGP 19, has some SBK circuits and some road tracks.
SBK 08/09 (PS2): Similar to MotoGP 07+08 and even playable with digital buttons and D-pad.
MotoGP '07 (XB360): unrealistic physics and AI, but plenty of fun. Has LAN.
Project Cars 3: The closest you'll get to a Shift/PGR/Driveclub sequal (EXPERIENCED assists).
Sega Rally: great physics, but too few stages, even with mirrored/reversed ones (avoid 4th stage).
MSR: an awesome 1-player game (MP full-screen time trial battles with the MX5 should be good).

Gran Turismo has become a sim with a static chase camera and although Forza has improved its track lineup over the years, the brake-distance in FM7 is still way too long.
(For FM7 use STCC/WRX-beatle, ABS on, steering norml, TC off, SC s-easy, pro skill, fric. off.)
Early versions of Gran Turismo 2 had a bug that prevented 100% completion. Mods like the A-Spec mod fixes some bugs and re-instates lost cars. GT4 has a Spec-II mod.
Grid Legends is too simplified with very short braking distances. You can have some fun with it tough in cars like the Shelby Cobra.
Shift 2 has mods (Patch 1.2 or 1.1) that makes it playable with a D-pad. Add the Gran Turismo View Mod to it.
Set the brightness in Project Cars 3 to 60 (has no split screen/LAN).
Xbox360 gamepads need a driver for vibration with some older PC games.

If you are not into regular circuit racing then you might like: Wreckfest CE, Forza Horizon 2, GRID 1, GTA 5 or Burnout 3.

Rally games:

RalliSport Challenge 2 (XBOX): this is what Sega Rally 2 should have been.
WRC 3 (PS2): 9 stages per rally and the cars have exciting arcade handling. Analogue buttons on.
WRC 4 (PS2): 6 stages per rally. You can turn off the evolutions to make it more challenging.
WRC Rally Evolved (PS2): the most realistic physics of this trio. Driver aids off, steering fast.
Dirt Rally: (no split screen/LAN): D-pad steering sensitivity / linearity: 100 / 0 (WRX 100 / 100).
WRC 8: The best KT game for digital buttons. VW Polo handling feels good. Steering sens. = -3.
Sega Rally (Saturn): the PC version lacks music and has compatibility issues + graphical bugs.

Alternatives:
WRC 9: More twitchy on tarmac than 8, but less than 10. Use WRC 8 sound. Steering sens. = 10.
EA Sports WRC 24: Can have stutter problems (due to graphics engine / anti-cheat) on PC.
Dirt Rally 2.0: The DiRT version of DiRT Rally ITO physics, but still better than any DiRT game.
                     D-pad steering sensitivity / linearity: tarmac 85 / 10, gravel 100 / 8.
WRC 10: More realistic than WRC 9, but you feel less connected to the car. Nice visually.
              D-pad steering sensitivity: tarmac = 7, gravel = 10. DLC not necessary. It has Greece.
Dirt 3CE: RX/H2H tracks + RX cars/Kadett only. Add Hulk remake+skins+RFPE D.Rally Physics.
Richard Burns Rally if you want a sim.
(Avoid WRC 2010 to 7 (rather get Seb Loeb Rally Evo), Rally Trophy and most Dirt games.)

F1 games:

F1 '05 (PS2): LAN and a harder difficulty level is what this game needed.
F1 Championship Edition (PS3): PS3 version of F1 '06 with LAN.
F1 2020 DSE: if only those track limits were accurate. Use the Ferrari 2004. Has LAN.
Assetto Corsa V1.16: use Ferrari F2004, skins, Flag-to-Flag physics, steering wheel or Xpadder.

Alternatives:
F1 2013 CE (PC): needs update 6 and a tire wear mod, recommended TrueAI/LegendsV2 mods.
F1 2014 (PC/PS3/X360): all 2013 bugs corrected. Has many historic season mods. Has LAN.

I still have to try:
Grand Prix Legends HD (PC)
Grand Prix 4 (PC)
F1 Challenge '99-2002 (PC only, PS2 version is a different game)

Good Steering Wheel games:

When it comes to steering wheels, the cheap ones just aren't worth it. You could rather just play with a controller. A good steering wheel needs to have resistance & vibration and some metal in its makeup. The stick shifts on more expensive ones are sometimes still not good enough.
I recommend turning up vibration and resistance levels to make the experience more immersive, to make up for the lack of g-force.

Gran Turismo 4 (PS2, has LAN)

Has no dash view, but the Spec-II mod fixes that and the chase cam height.

Assetto Corsa Competizione (PC, has PVP)
A GT3 sim. It is like a modern day TOCA in terms of physics, where the car will lose traction and swop ends if you go too fast into a corner. You can use Xpadder for D-pad steering.

Assetto Corsa Ultimate Edition v1.16.4 (PC, doesn't have LAN)

Use only the Ferrari F2004, all add-on F1 tracks, 2004 skins, Flag-to-Flag physics.

www.racedepartment.com/downloads/assetto-corsa-the-f1-game.19922/

Dirt Rally v1.23 (PC, doesn't have LAN)

https://www.racedepartment.com/downloads/dirt-rally-mods-package.20098/

Use classic Mini RX.

Linearity = 0, Stability control = 5 (off for classic mini RX).

Settings for controls/wheels have to be redone each time after they are switched.

The moon-physics fix isn't online compatible.

Grid Autosport Complete Edition (PC, doesn't have LAN)

Cockpit cam no blur mod and mirrors/betterRflex by Mortal mod.

Gamma 1.00 + Hulk's lighting mod 1.1.

Tag & flag mod.

No dirt mod (on wheels) mod.

Wheel weight 100% (steering linearity 0, -10 for digital gamepad).

Turn traction control off.

Cam mod 1.4 no gear jolt, no pull away, no shake fx, chase cam height 2.2.

Gran Turismo 6 Update 1.22 (PS3, doesn't have LAN)

Use the GTR Nismo GT3. Updates include Red Bull Ring and Midfield Raceway.
For wheel use: active steering off, tractionC 0, active stability off, ABS 01, skid recovery off, sensitivity 0, cockpit cam.
For D-pad use: active steering off, tractionC 0, active stability off, ABS 10, skid recovery on, sensitivity 7, chase cam.
The D-pad settings takes it away from being a sim and makes it feel more like a regular game.
Set exposure to +0,7.

Race Injection (Race 07 / GTR Evo, Race On) V1.2.1.10 BTCC Edition (PC, has LAN):

Super Touring Seasons2.11 (changes a mediocre game into a good game)

STCC - The Game (for sound)

Porsche Mobil1 Supercup v2.5 by ZeZeZoom and kiwisteve

WRC Toyota Corolla, Audi RS4, F3000 skins '93

BTCC TRACKS: BritishGP Int, Croft07, Knockhill 05, Oulton 05, Snetterton 05, Thruxton_ST

Nurburg (GTR Evo, need sky textures), Laguna Seca + Road America (Race On, need adboards)

Barcelona 03, Donington 04, Enna 03, Zhuhai 04, Spa 03 (GTR2)

ADD-ON TRACKS: 2004Lemans, Adelaide2011, Bahrain, Barcelona07, Chang, Cota, GP200, Hungaroring 2011, Indianapolis_IRC, interlagos v2.0, MoscowRaceway 2013, PaulRicard_HTTT, Sepang v2.0, Singapore08, Suzuka2012

Real Road Effects v1.40 by Denis Vorobjov + Stunning Rainfx new v1.1 v1_1 by splitpbt

brighter headlights For Race07 Cars and AI + Flameout GFX v2_1 by Prodrive

Grid Order Background v1_0 by jason_peters + New Sky Textures GTR Evo v1.0 by batito

Clear visor + TV cam for open wheelers

(Xbox360 controllers have a bug when using digital steering, left can stop working)

Optional:

GT Pro, GT Sport, GT Club (GTR Evo), Muscle Club (Race On) cars

GTR2 cars (change IDs 110 to 255 with STS v.2.05), DTM '94 mod (use swingman view)

BPR F355 Challenge Mod v1.1 by Wildman_fr & Tommi-TAG

FVR V8SC2011 Ausie V8 Touring Cars v1.24 by FVRFactor - Gunthar Rowe

Worth a try:

F1 2013 CE classic cars with cockpit view.

F1 2014 with behind helmet view.

I still have to try:

Richard Burns Rally: has many mods.
R-Factor 2 (PC): Very realistic physics, FFB and AI, but limited official content.
GTR2 (PC): Not made for use with controllers. Has LAN.
Forza 4CE & 7UE: FM7 on PC has compatibility with many steering wheels (no split-screen).
Gran Turismo Sport
Dirt Rally 2.0 GOTYE & EA Sports WRC 24
WRC 8, 9 & 10
Dirt 3CE
Project CARS GOTYE & Project Cars 3 DE
F1 2020 DSE & F1 25
Automobilista 2 (the Project CARS 2 successor)
Shift 2 Unleashed
Wreckfest CE

Bike games:

PGR4 (Xbox360) Use furthest 3rd person view.
Solid, simplified handling with detailed bike animations.
Has LAN capability.

MotoGP 19 Retail Release Version (PC/PS4/XB1)
The best MotoGP game, but it has no split screen/LAN. You can use MotoGP 17 for split screen or '07 for LAN. MotoGP 25 has split screen and arcade mode handling.
For a couple of years Milestone made bike games that are easier to play. It can even be played with a d-pad / keyboard, but you have to play the games un-updated. You have to play the games that are on the discs (not the day one STEAM versions). They are called baseline versions. Any updates afterwards just made the handling worse.
The leaning speed and braking distances (in MotoGP 19) are more realistic compared to that of its updates and the bike has a feeling of weight to it. The off-track grip is on a Forza Horizon level though (compared to MotoGP 20) and the game will have some bugs, like your nationality that can't be changed.
You can install the game offline with the disc to get this version on the main consoles. Just don't ever update it. On PC, however, you have to use the first manifest file and a program to extract the original game from the discs (+ you have to add one file). Then play it in steam offline mode, launching it directly from the game folder as admin (it won't start if you click play in Steam).
After writing this, I played the fully updated game briefly and the leaning speed seemed the same so maybe they brought it back towards the last updates. The bike behavior on the curbs was still the same though.
[MotoGP 20 is the same story... get the disc version. Compared to 19, the off-track physics (not talking about rag dolls here) and AI are better, but the handling doesn't have that sense of weight to it and the braking distances are shorter. You can update it if you like simulations.]

RIDE 3 Retail Release Version (PS4/XB1)
This can be used as the unofficial SBK game. You can set the physics at normal and the traction control at high (low for fully updated version), with brute force buttons. It will give you a good balance between traction and sliding, which you don't get in the final version.
Things like Macau daytime won't be in the game, because they continued to work on it for a year after release, but all DLC packs are valid with the disc version (although some bikes might be excluded). This game is a good companion to MotoGP 19 and has PVP.
The earliest version on PC is the Day 1 Steam version, which already has differences compared to the baseline game. It is still similar enough, in terms of physics, to the baseline game and can be updated to update 3 if you want to. It can be used in Steam offline, but you have to swop the bad files for the legit files in the last version, if you know what I mean.
You can use RIDE 2 for split screen with the MotoGP 16 offline circuits mod added.

SBK 09 (PS2 only, the Xbox360 version is a different game)
Basically the same as SBK 07 & 08, but it has one more circuit to give a total of 3 new circuits. It doesn't have Silverstone from 07, Brands Hatch from 07/08 or Vallelunga from 08. Instead you get Kyalami, Miller Motorsports Park and Portimao. It has split screen.
Use anti-lock brakes, brake power 100, joint brakes, no traction control, unrealistic inertia and no rider help.

MotoGP '07  v1.1 (PC/Xbox360) Use 2nd furthest chase view.
Good handling/graphics/gameplay, but AI racers sometimes too aggressive when overtaking.
Turn off crowds for PC version (has Windows7 compatibility issues = crowds flicker).
Run MotoGP.exe in Windows2000 compatibility mode.
Unofficial physics patches exist (MGP07 Physics V2.5 + Default), but I don't recommend them.
Has LAN capability.
XBox360 update not available.
The AI in the PS2 version uses much better racing lines (it is made by the '08 developer).

MXGP 2019 Retail Release Version (PC/PS4/XB1)
This is the gateway game for car racers who want to get into dirt bike games. You can just focus on braking/acceleration points and the racing line, without having to spend lots of time in the air or know how to do a rhythm section (use standard physics). It has PVP. The difference in physics between the first and last versions are minor.
The people who are into MX/SX games will prefer the fully updated MXGP 2020 or MONSTER Energy Supercross 2 or 3.
I don't like the unrealistic MX vs ATV games, but Alive will be your best bet there, although the fans will prefer REFLEX.

MotoGP 08  v1.1 (PC) Use 2nd furthest 3rd person view with advanced handling.
Has underdeveloped bike-on-bike contact physics and no split-screen.
PS3 version has slow frame-rate, delays on sound effects, dull graphics and steep digital button acceleration curve.
Run Launcher.exe in Windows2000 (ME for Win10) compatibility mode with no admin rights.
Multi-player is online only.
(MotoGP 08 on PS2 has crowded AI racing, lots of wheel-spin and multi-player is split-screen only with slowdown).

MotoGP URT 2 (PC/Xbox) Use 2nd furthest 3rd person view.
Simple/fun arcade racer.
Not to be confused with Namco version (MotoGP 4 is the best Namco version).
INSTALL "tsp_codec_install" AND “Microsoft_DirectInput_Mapper” for Xbox360 controller.
Run MotoGP.exe AND Launcher.exe in Windows2000 compatibility mode.
Use the 2016 patch (use the 2015 patch for URT3).
Has LAN capability.

Tourist Trophy is a scaled down GT4 and has no LAN. MotoGP 4 feels slow and unfortunately SBK 08 on PC has bugs.
If you want a sim: Tourist Trophy, Superbike 2001, GP500 (2012 mod), GP / MX Bikes and MotoGP 24.
(Avoid: MotoGP 09/10)