I can't really do any of the quests because I don't qualify for any of them. And I can only do exploring for a little while until my pets get tired. IS there other ways for me to get money?

Breeding and selling pets or sending pets to the pound.

Selling items and materials you get from islands.

There is a thread (I forget where exsactly) where you can borrow pets for quests.

On the News page on the right you can vote for beastkeeper and get 1500 everyday.

If I am missing something im sure someone after me can help.

What I do sometimes is either adopt a random pound pet then sell it (if someone wants it), or you can find good cheap pets in auctions.  After the auction time you are stuck with a good pet, you can sell it for more than you got it for.