Remember that you can check the replays of races to see what tires the AI are running, since it doesn't tell you beforehand. This can save you some frustration, and also explain why you might have blown them away worse than you expected. They always seem to stay on the same type of tire throughout an entire set of races - meaning for example if the first race of "clubman cup" they are on comfort softs, then they will be for all the races in clubman cup. Tires are one of your most effective means of controlling the game's difficulty, so don't be afraid to change them up or down as needed.
I'd recommend against going straight to racing soft tires on every car, as it can be a tough habit to break. Try not to think of different tires as "better" or "crappier" than others because they really aren't, they're just different. Sports and Comfort tires have less grip than racing tires but tend to allow more sliding before completely losing control.
