If you already have the car you want to race, get in it. For my race, I've chosen the 1967 Eagle-Westlake T1G. Then go to Single Player > Free Play > Setup > Race Setup. From here you can select the track (in my case, Spa); day, night or rain (if the track offers those options); number of laps; the number of drivatars; and the car division. Make sure under game type, you select Circuit Race. This is where you choose between a regular race, timed race, drag race, drift race, etc. When you've got it how you want it, then press A to accept the changes and go race. When the race is over, it will automatically send you to a different race track. If it's not where you want to go, just go back to the setup screen and change it. If the game insists you homologate the car to race it and you don't want to, go back to setup > advanced setup and change "allow upgrades" to "no".
On big bonus is that you can race any (non-DLC) car in the game, even if you don't own it. Under race setup, set the car division to "any", accept, then select car. From there, you have three options: race a car you already own, buy a car and race it, or rent a car for free (but you won't earn any XP or credits, and you can't modify the car).