You may buy as many different types of tires as you want, but you only need to buy each type once. The reason they are so expensive is that you get a fresh set every time you pit, automatically.
Before the race starts (or during Pits, I guess) you can choose the type of tire to run (assuming you've already bought that type) in the Settings menu. Click on the red toolbox. You can also get to the settings menu from your main garage page by clicking the Start button on your controller.