It helps a lot to switch to the first Morty you will need *before* entering the ring. For at least the first battle or two you can see your initial opponent standing directly behind Jerry. So if it's say, a paper type, you will want to switch to a scissors Morty before the match begins. That will save you a turn. If you don't know how to do that, use the device icon in the upper right (forget what that thing is called), tap "Your Mortys" and then you can switch their starting positions. Top one goes first.
Also try to heal your Mortys in between matches instead of during, unless they are at critical HP.
Hit your opponent right out of the gate and keep hitting. Don't take time to use buffs or debuffs. Just knock those suckers out as fast as possible.
FWIW, I think I used sausage, boulder, and goo/blob/whatever Mortys.