Noob here

So a couple things, what are the purple diamonds on my pets that say #/#? Also how do I get more slots?

You buy permits for pet slots in the shop, it's like 2550 coins, the blue numbers are like which pet was made first, for example, there's like 1200 pets in the game right now, one of your pets is the 804th pet made, now the purple diamonds are showing how much the same kind pet you have are made, for example on your shark it says 1/3, it means there's 2 other pets of the same one you have, hope this helps!