I like team Kicker the most. Unlocking Teams works like this:

You have to be really good for Team Kicker to unlock. How hard its for you depends on the Tier from your Team. Better Teams needs more time to unlock.

Repeat the offer on the left (Series) with the 2 Events of the Car class in which you want to unlock the request Team (the "C" Class, for example).

Then you unlock, if you drive well, another Team. This is usually a little better than the Previous one, but does not always work if the Team Driver is bad and your Team Rank is worse after the Event. So always take the Team with the strongest Team Driver. Do this as often until you have a Team that is strong enough that you can achieve Team Rank 1.

Then comes a random strong Team that makes an offer. Just continue to take the strongest Team until the desired Team appears. Always repeat the same short event, at some point your Favorite Team will appear.

Unlocking a Stronger Team (as you can see in the Tuning Skills: Low, Medium, or High) you had before, going up an "Tier", will also affect the International and Championship Events. You also unlock new offers there.

Everything under Championship has an impact within the Car Class. The Championship also influences the first Event of the next Class. Then you can play the unlocked Team if you want and continue playing it upwards in this Car Class.

Unfortunately, it is not sure that the same Team in the next higher event makes an offer again. Maybe you have to Grind there again the shortest Event until the Desired Team makes an offer.

If you have 2 Teams to choose one with Tier 0, one with Tier 1 (The Teams Tier level is based on the Tuning level (none=0, low=1, medium=2, high=3), you have to take the higher Tier Team, as the game does not remove a higher Tier Team.

Always you drive the Team with the lower Tier, the game keeps the other Team and does not replace it with a random newer Team (Ravenwest never disappears and when you play Ravenwest you can not unlock a new Team).