There are some quests that allow you to collect gems without paying. First, though, I advise that anyone starting this game saves as many gems as possible from the very start. Quests will need you to spend gems to obtain animals.
So, throughout the first few levels, you should receive the opportunity to receive 2-3 gems for completing a quest. SAVE THESE GEMS! From those levels to Level 20, I never received any quests which rewarded gems. However, do not spend real money buying gems, even though the temptation is strong (I almost did it myself).
Once on Level 20, you should receive a quest which you have to a) buy a for sale sign, b) obtain 10 five-star ratings, and c) have 20 neighbors. This quest awards you 5 gems on completion, which I just added to my fund and bought a St. Bernard.
I'm on Level 20 now, so I don't know about any quests past Level 20. If anyone else does, please post! Thanks!