Do the different gems do anything?

Nope. Their only purpose is for turning your pet(s) galactic.

How does that work?

At the Coda Caves you can ask to Bless Your Pet and Itallo will list some gems. If you bring the ones/colors he asks for your pet will turn galactic. Only on days the caves are open though.

Ok thank you :)

Remember the caves are only open on the 6th and 7th day of every month :)

Thank you :D I got my pet turned :)