The easiest way to get diamonds is to buy them from the App Store

however if you don't want to buy them there are two main ways you can earn Diamonds without having to pay for them:

If you are a loyal player and play every day you will be rewarded by the Daily Login Reward. By playing every day, you can earn 70 Diamonds per week and 19,000 Coins per week 

You can optimize how many coins you earn by wagering before each race and collecting all your coins daily from your sponsors and watching the ads at the end of each race.