Here's some things you ought to know first:
Each Faerie asks for something different and grants different improvements:
Air Faerie: wants Grooming items, increases Movement
Dark Faerie: wants Toys, increases Health Points
Water Faerie: wants Books, increases Defense
Fire Faerie: wants Clothes, increases Strength
Light Faerie: wants Trading Cards, increases Levels
Earth Faerie: wants Magic Items, fills your pet with food - POINTLESS! Dont do these quests!
Faerie Queen can ask for anything and everything (usually rare items) and either makes your pet Super Strong, raises a Level or increases Health Points.

When questing you cannot use the shop wizard, so your options are:

1) Check out the auctions
2) Check out the Trading Post
3) Browse through random shops
4) Ask your friends or post for help on message boards
(other people can search the shop wizard for you)
5) Refuse the request - there is NO penalty, other than not being able to accept another quest for about 10 minutes.

When you finally have the desired item take it to the quests page and push the button that says, "I have the item!"

At this point, the item will disappear from your inventory, and the faerie will give your pet her special gift. You will NOT get your item back.

Questing Tips

1) Use the item before you lose it! If you had to go find a grooming item, or a toy, take a second to groom or play with your pet using the item before you go back the quest page. (I even look at the trading cards to see if the pictures are cute :) You get a happier pet and the faeries can't even tell if their item was used.
2)Don't be afraid to REFUSE REQUESTS. The faeries sometimes ask for very rare, very expensive items, and it's not worth blowing your bankroll for a Health Point or even a level. Which brings me to...

3)Don't bother with Earth Faerie quests. Why would I spend thousands of NP on a magic item just so this flaky faerie can stuff my pets with food? I can do that myself for under 30 NP. Always walk away from this faerie!

4) Decide how important questing is to you. Do you want to raise your pets' levels fast, or is the Battledome just not that big a deal? Either way is just dandy, of course, but if you really want to race through the levels...

5)Quest for one pet at a time. If you have 4 pets, you never know who's going to get the benefit of the quest. If having a super-powerful pet is your main goal, get one pet and go for it!

Lastly, if you're just starting NeoPets, don't be in a hurry to quest! Take your time to enjoy the game, play with it, build up enough NeoPoints to keep your pet full of food and comfortably well off before you start questing.