As someone who's new to the game I'll post here on other championships when I find them but if you go with these cars for the starting tournament (both parts & rival race) & the off road tournament you'll breeze through it without modifiying your car. 

Lotus (starting car of the 3) 
then 2nd championship in that class go after the Lancier Delta 
 
For off road you might like the hummer as it's cheaper & needed for a photo on hawaii though go after the land rover. Haven't done the rival race off road yet but aceing all the tournament bits without much problem. 
 
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Okay whoever decided on the design of this game made some things quite hidden in the interface. As for selling your cars, aparently you can't sell the rewarded cars but anything you buy from the dealers can be sold if you do this. Walk into a dealership & go to buy a car that's a little above what you have for $, there'll be a prompt to sell a car to make up the difference.  
 
Took forever to figure that out after buying a car intially from the used car lot & wanting to buy the new one for the interior & other options only to find that instead of overwriting the car you had or making up the difference if you had the car but want some of the perks from the dealership version but nope, you just wind up with two of them. Easy way to tell them apart is when you go to compare or sell them there should be a mileage marker in the lower corner of the car so it's easy to see which one is new or used based off that (if dealing with what I did where you had one new & one used car without mods).