I'm faced with surrendering beloved pets on a regular basis - I work at a kennel where we often host the overflow from local rescues, and I am a pet foster parent. There are only a few dogs that I didn't become attached to in some small way, and one or two I really hated to let go of, but I know I can't keep them all.
If you are faced with giving up a pet and there are legitimate reasons for doing so, make sure you take every precaution to ensure finding your pet a happy, healthy, forever-home.
1) Exhaust your options. Before you make your final decision, make sure you have exhausted all of your options for two reasons. First, there are hundreds of thousand of unwanted animals out there looking for homes, and unless your pet is very lucky, or you have the luxury of time or good connections, chances are your pet will not find a new home if taken to a shelter. Second, if you make a decision in haste, you will regret it all of your life.
There are many reasons for giving up a pet - finances, time, moving, conflicts with other pets, allergies - but if you do a bit of research and a bit of re-prioritizing, you may find an alternative that will allow you to keep them.
2) Prepare Your Pet. A well-behaved, well-groomed, healthy and "fixed" pet has a much better chance of being adopted. This is important to note if you're surrendering your pet because of behavior problems! Most "normal" people don't want a pet that is going tear their house apart, barks uncontrollably, refuses to use the litter box, or comes with fleas or a litter of puppies on the way. By "normal" I mean your typical pet lover - some one looking for a pet and not dog fight bait, puppy mill breeding stock, or another animal for the "collection".
3) Good Pictures and Description. A picture is worth a thousand words. Pets that look cute or "needy" in their pictures do better than pets who appear as a blur or in mid-snarl. And try to avoid the demonic red or white eyes - take the picture outside on a slightly overcast day for best effect.
Your description should be brief and accurate. If your dog has issues, don't lie about them - you don't have to highlight them, but don't mislead prospective adopters. Be honest about the reason you are surrendering them. And include your phone number and email address!
4) Your Own Contacts Talk to your own animal-world contacts, and ask if you can post notices at the vet's office, groomer's, the local pet or feed store and kennels. Most will be happy to help you locate prospective adopters, but you still need to do the leg-work yourself. They may even have tips on how you can keep your pet, or tips on how to improve your pet's chances of finding a forever-home.
Family members can be a good option, but don't assume that a family member will give your pet the home you expect. A friend was forced to surrender her dogs, and thought her cousin would provide a good home. She discovered a few weeks later that the cousin decided to "help" one of the dogs conquer his fear of the vacuum cleaner by tying him to it as she worked - the dog was totally traumatized. Fortunately, the dog was returned to his original owner, and when we last spoke, she was reconsidering her options.
5) Work With a GOOD Rescue Group. Contact a local rescue group with a good reputation. Chances are they won't have room to take your pet immediately, but they can help you to get started with the adoption process. Petfinders and other similar sites only list animals with registered groups, and on-line adoption sites are the best way of finding a good home for your pet.
If you have a pure-bred dog, contact local breeders (certainly the breeder who sold you the pet - unless you've discovered they are of the backyard variety) AND contact the appropriate breed rescue group. Again, they may not be able to take your pet immediately, but they will have contacts.
6) "Professional" Adoption. Every good rescue group has a process to ensure their animals go to the best homes possible. The process generally includes legally binding pre-adoption forms and adoption contracts, visits to the prospective adopter's home, reference checks and especially a call to the adopter's veterinary office. A post-adoption visit may be required too.
Even if you don't work through a group, consider creating your own formal adoption procedure. Anyone who wants a pet for legitimate, loving reasons will have no problem with paperwork or reference checks. If someone hesitates or becomes irate at the request, they probably aren't going to be a suitable pet parent.
There are forms available on the Internet through various rescue groups, Google "pet adoption agreement". You need to tailor the documents to your circumstances and make sure you keep a copy for yourself after the adoption.
And do the background checks! Speaking with a local animal control officer about why they no longer allowed pit bulls to be adopted, he said they stopped after a dog they had rescued was found dead at a raid on a dog fight - he had been used as bait. A cursory background check had been performed, but had they dug deeper, the dog would have been saved a horrific end.
And don't ever surrender a pet to a stranger without checking them out, no matter how kindly in appearance and no matter what the sob story.
7) "Free to a Good Home. "Free Kittens" or "Free to a Good Home" ads are a REALLY bad idea - they don't always attract responsible individuals. Your "adoption fee" should be low enough that it isn't a blatant attempt at money-making, but high enough that only someone looking for a companion will consider adopting. A good rule of thumb is to ask for the cost of your pet's spay/neuter surgery, plus the year's inoculations.
A valid reason for "waiving" the adoption fee might be if a family member or friend is willing to adopt your pet (especially if you can make sure the animal is happy) or if an elderly person is considering adopting an older pet (quite possibly a match made in heaven).
8) Handing Your Pet Over. Make sure all your pet's records are available - shot records, spay/neuter certificate, licenses. Make brief notes on your pet's likes and dislikes, and any special medical or dietary requirements.
When the time comes, let the new owner take the pet from you, rather than you leaving the pet. In theory, this eases the transition, because you aren't "deserting" them - we use this strategy at the kennel to avoid separation anxiety, and while it isn't foolproof, it does help with some animals.
And then step back and let them get on with it - your pet is no longer yours. I usually wait two or three weeks, then call or email, just to make sure everything is well. And I always offer to be available for consultation if there are problems or to take the animal back, at any time.
9) Euthanasia. If your pet is very elderly or has serious or difficult health problems and you cannot find a suitable home, you need to consider euthanasia - especially if the only other option is to surrender them to a shelter. It will be kinder to your pet in the long run, sparing them the trauma of separation and time in a stressful shelter. They will leave this life much more peacefully if their beloved human is there at the end.
And making sure any pet ends their days with a minimum of suffering is EVERY pet owner's responsibility.
Being forced to surrender a beloved pet is every responsible pet owner's nightmare. Consider your options very carefully before you make your decision, then take as much time and make as much effort, as possible to find them the best home possible.