Many people think adopting a pet is a great thing, and indeed it is. Knowing what to expect before hand can make the whole process easier. Let us make one thing clear, adoption is from shelters, purchase is from stores or private individuals.
Call the shelter before hand. Specifically you need to ask them when their adoption hours are. If adoption hours close at 5:00 pm you will want to be there at least 1 hour earlier to have a good look at the available pets.
Ask the shelter what you need to bring. Generally they want picture identification, proof you are over 18 years of age. If you rent they also need landlord permission to have a pet, so either bring your contract, or the landlords phone number. They may want to contact your veterinarian, if you have one.
You may wish to inquire what their adoption fee is and what forms of payment they take. Adopting a pet is not free. Most shelters are non-profit and poorly funded. They have to pay for the vaccinations, deworming, used English saddles for sale online, and so forth on the pets. Your adoption fee helps cover this and is usually lower than if you were to take a pet to the vet and have this done on your own.
If you are adopting a cat, or small dog, bringing a carrier, or box is a good idea too.
When you arrive at the shelter talk to somebody and let them know what kind of animal you are looking for. Be specific, do you want a cat that is good with dogs, a dog that is good with kids, or a rabbit that comes with its own cage? What age, gender, etc. of pet are you looking for? This will help the staff point you in the right direction.
When you find a pet you might like ask the staff if you can take it out of its cage, or in the case of dogs, take it for a walk.
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When you make up your mind ask the staff for an adoption questionnaire, or whatever they do to complete the adoption process. You may want to have them hold the animal for 24 hours while you get supplies (do not buy food before you buy your pet, you should stay on the same kind of food the shelter is feeding for a few days and change it gradually later).
If you do not find a pet to your liking ask if they have a waiting list. This is great when looking for a specific type of pet or breed. Ideally the shelter will call you when a pet comes in.
Keep in touch with the shelter after your adoption. Most are there to help you if you have any questions or problems later on. They will also want to know when you spay or neuter the pet if that is a requirement of the adoption.
Remember most shelters have contracts that if at any time you have a problem and cannot keep the pet, they will take it back.
Good luck with your new pet, and thank you for saving a life!