How do I raise my pets health, and other stats so that I can battle them and not die on the first turn?

Go here https://subeta.net/explore/train.php :3

Train all your other stats (strength, health, defense) until only level is left, then start again. Level doesn't do anything except determine how high your other stats go, and if you train it too much first the time a lesson takes AND the expense goes up before you've gotten the best value for your buck.

You can train your pets here. (Riverside map, sort of in the middle)

There is also a Beginner's guide in the battling area to get you started.