The best way to get free diamonds in Magikarp Jump is to complete achievements in the game. Checking out the achievements page now and again is a good idea as sometimes you have unlocked achievements without even knowing about it. The achievements page is got to via the Menu link.
You can also get free diamonds via completing League battles and by random encounters.
However, if you are paying to play the game you can unlock the Diamond Miner and get 100 free diamonds daily! Check out our page on the Diamond Miner to find out how to do this.