I was battling when my level 10 pet died. The only other pet was a level 0 pet, so I went back to battle. The bandit I was battling was level 12. Almost immediately, that pet died, too. How do I level up my level 0 pets without killing them?

It's easier when you have two or more party slots. If you don't, take them to the lowest lvl area possible and hope for the best. Or use an exp orb if you have one.

They lvl through multiple ways:
Battle-easiest and cheapest way to gain lvls try lvling in lower lvl areas for lower lvls pets.
Orbs-You can buy these in trade brokers or gain from missions or battles. Gains exp instantly, but can get pricey.
Missions-Min to do the lowest lvl missions is an hour. This is not only a free way but you gain items, and gold for this.

If you die you can go to town, go to the plaza, then go to the healer. they will heal you for a price.