I generated a new pet that I want to be on my battle team, but it's difficult to train because if it dies in battle, which it usually does, it won't get any experience. Is there any way to level a new pet with your lvl 60 team effectively? The experience orbs are too expensive for how many I'd need. Also my tank has earth for the taunt, but it doesn't always work.

Personally, what I do for leveling new pets, is I put the new one in a team with level 60's and take them out to say, Trollsblight Caverns, fight there for a while, that'll bring them up in level quite a bit. Basically just go in order along the sections and they level up fairly fast that way.

What AaronAmpora said. Put them in a level 60 team, and fight in a mid-range level place, then slowly go up. Also, use that pet to go on missions, especially if you're going to be offline for a while (eg sleeping). It won't rake in as much money, but it's a good way to give pets xp without doing much.

I usually put pets that I want to train with a lvl 60 light elemental pet so it can heal them with "Fairy Light" when needed. I choose "Faerina Shrine" as my first place to train them because I can finish a quest while leveling them at the same time. It's like killing two bird with one stone! :D
After that, I'll finish other quests. When I've finished the quest in Aryth Spire, my pets' lvl are usually about 35 so I can just let them doing missions until they reach lvl 60. :)