When you play the story mode (Team Adventure Mode) in Team Sonic Racing, you can only play as team Sonic, which means that you can only use Sonic, Tails and Knuckles. However, you might want to play as the other teams as well, but how do you unlock more characters in Team Sonic Racing?
The answer is simple: just play the Team Adventure Mode, and you will unlock the other teams as you complete new zones. Do this and you will unlock Team Rose first, followed by Team Vector, Team Dark, and finally Team Eggman. It takes a little while to reach the point where you will unlock all the teams, but just keep doing the adventure mode, and you get them eventually.
There are a total of 5 teams and 15 playable characters, and you get all of these by doing the main story mode. So just enjoy the game, and you will gradually unlock them all.
Luckily for us, you don't need to do anything to unlock them for multiplayer, neither online nor locally. So in these modes, feel free to use whichever team you like the most, even if you haven't unlocked them in the Team Adventure Mode yet.