You can avoid the other Morty trainers by preventing them from spotting you. Make sure to use the environment to your advantage if you're not looking for a fight. Just be sure to mind their field of vision. It can be helpful to go to an area's Rick and take him out first, so that then you have a portal to use when you're ready to return to the Citadel.
Diversify
You should try to have a Rock, Paper, and Scissors type of Morty in your lineup, so you can successfully counter attacks from other Morty you might meet. Remember: rock beats scissors, paper beats rock, and scissors beat paper.
Don't Neglect Improvements
You can improve your Mortys by combining two of the same type, or by feeding them seeds. Don't forget to take these steps, since the result will be a stronger team that is better prepared to take on the various enemies you'll encounter.
Prioritize Badges
With badges, you can buy stuff from Salesman Rick, including Control Chips that you'll need to gather a proper stable of Mortys. For that reason, you should prioritize earning badges, even early in the game.
Sneaky Training
The Mortys that you let participate in fights are the ones that earn experience points. Sometimes, you'll want to power up a weak Morty that doesn't really stand a chance against current opponents. Begin by having your bruiser do most of the required damage, then switch out Mortys at the last minute so the rookie gets the experience boost at the end.