Basically the levels just go numericly up to the highest of 48 (I think). But there are different "Badges" that you earn and can "wear." Under your name it will show either your current level and school, like you are an Apprentice Pyromancer and will advance to Journeyman Pyromancer next. I am a Magus, but I think the top level is Grandmaster. Or you can choose to wear your "Badge" of "Three Streets Savior" or "Golem Tower Champion." You get more badges by defeating more of each type of enemy.
As for the pets, some of the Bosses "drop" them, like the snow serpent and dark fairy. Someone told me that fighting the Gobbler King's minions got them a snow serpent, but I fought those guys repeatedly and never got one. Maybe you have to be Ice School. I did get a dark fairy, but I like my Death Leprechaun better because he adds a card to my deck. He only cost 21,000 gold. Each world has someplace you can buy pets. Hopefully they will be able to do more in the future.
There are fansites that have guides and stuff, just like Stratics. I signed up on Wizard101central.com because it seemed pretty well organized and I found some good info. Lots of kids post there though, so some of the posts are a little wacky.
Anyways, you know you can share stuff that you get but cannot use with the other characters on your account? There is a Shared Bank which is like gifting to the other character. So if you get ice stuff and your daughter's wizard is ice, you can put it in the shared bank and her character can get it out. I thought that was a pretty clever idea. Players can't gift each other though, only trade treasure cards.
Sorry if this is too much information, but I get carried away I guess.