UPDATE
The Neopets Pound has been down for months at this point. Nobody knows if this method will still work when it opens again. We will re-evaluate and let you know if anything changes. Since the pound is DOWN you can only place your pets up for adoption and nobody knows if you will be able to recover them at a later date. If you do not want to lose the pet forever, we suggest holding on to the pet.
-NeopetsGuide Team
Bye Pet! Hello New Owner!
Are you wondering how to tranfer a pet to a different account, or to the new owner of your pet? If so, you have come to the right place. 2 words, The Pound. The only way to get a pet to another account, or to the new owner is by tranfering the pet useing the pound. But don't worry, you don't have to use the adopt search engine. There is a safer way. Before you use this method double check that the other account is old enough to adopt your pet. (4+ months for limited edition/expensive paints) This is the here is a quick adopt link:
http://www.neopets.com/process_adopt.phtml?pet_name=petsnamegoeshere
The Quick Adopt Link is a easy way to tranfer a pet safely in the pound. You put the pets name where it says petsnamegoeshere, but make sure you have enough np. By finding out how many np you need to adopt, deposit all your np in the bank, then go to the quick adopt link for that pet. it should say something like "you need ....np to adopt" so make sure you have that much np out. Remember that you can do this BEFORE a pet is adandoned. When you go back to the quick adopt link, it should say something like "sorry, this pet already has a owner and is very happy!" Thats means you have enough np out now, and the pet is not in the pound. Now, here are some sections useing the quick adopt link for different things:

Tranfering your pet to another account by yourself:
There is only one safe way (and sometimes that can go wrong) and that is to have two different browswers. Firefox and IE most people have. If you do not have FireFox, I recommend getting it. That way you can be logged into the account you are abandoning from and the account you will be adopting from.
Some people say it is okay to do it by yourself by logging out and quickly into another account. I DO NOT recommend this.

Tranfering your pet with another person:
If you are giving your pet to another person on neopets, or adopting a pet from someone, it is the same drill. But it is like you are sharing the effort. First, you 2 must have a set time to adopt. which is the time the person who is pounding, pounds. remember, if you just decided to give the pet to a person, and you want to give the pet to them straight away, give them enough time to get ready and get np out. Just say the time was set at 11:30pm NST.

Person Adopting
The person adopting should go there at 11:25-11:28pm NST to get ready with 2 windows open, 1 of them on neopets with the clock and another one at the Quick Adopt Link. Keep looking at the neopets clock until about 10 seconds before the pounding time, then go to the quick adopt link and keep pressing F5(refresh) for about 30 seconds. If it takes you to your pets Reference, the tranfer is safe. otherwise, someone else must have gotten the pet, the owner pounded late, or the owner didn't pound at all. Everyone has the right not the pound the pet if they changed their mind.
Person Pounding
If you are pounding, then you would also have 2 windows open. 1 of them so you can see the neopets clock, and the other at the pound. so you can pound straight at the time you are meant to pound, click on the pet you want to pound with all the messages. and stop at the last message which is "oh please noooooooooooo!!!" keep looking at the neopets clock, then at the time you are meant to pound, click on the oh please nooooooo and the pet should have been pounded. Remember to have 250np out to pound. if your internet is slow, I suggest you pound a few seconds earlier since it would take time to load the page.
Afterword
I hope you have a smooth pet transfer!
-Vivienne