While the elitist folks at the Cat Fancier's Association, American Kennel Club, and your local pet store may imply otherwise - there is no reason to go into debt for a loyal companion animal. Pets are given away for free or limited cost all the time. Hell, there was even a huge flap with the animal rights crowd 40 years back when the Empire was giving away Buffalo!
But, be aware....
That which was small and cute can get big and not-so cute. An untrained animal can be a curse, leaving you with a pad reeking of urine and feces with destroyed carpet and furnishings in short order. You may find your choices of housing being extremely limited or be extorted by greedy landlords for the right to keep Fluffy. Few things in life are as sad as having to give up an animal because of being unable to care for it. Be sure you can actually feed, train, have an appropriate environment for, interact with, and otherwise take care of a pet. If you honestly can not do this, let someone more stable have the opportunity to get the pet. Maybe later on in life when you are more stable, you can consider this again.
But, it is not all bad. A pet can also be a good friend who is happy to see you come home no matter what is going on bad in life. It has been scientifically proven that those with pets are less prone to mental illnesses like depression.
Check animal shelters, SPCAs, and Humane Societies. It would be in your (and the pet's) best intrerest to try the animal shelter first, as they still often put animals to sleep while the latter two avoid doing so. With vaccinations, sterilization, microchips, and ID tags, you can get a pet anywhere for free or reduced cost (almost always under $50). Be forewarned, many of these places do a brief inspection of the animal's future home and owner, so clean the place out, shove your brothers and sisters in the basement, and take the bong off the coffee table.
Craigslist and local newspapers always have pets for free/reduced cost. Free pets won't be damaged by genital removal or dribble problems from a botched spaying that the In-Humane society requires to avoid the pet death sentence. Un-mutilated pets are often more hyper and spirited, much like un-drugged hormone-charged ADHD teens....
Food and Board
In terms of paying for shelter or food for your pet, check to see what your animal can/cannot eat and in what proportions. You can often make a nutritious meal for your animal using household food stuffs for a fraction of the price of PetsMart's. Most animals can have a comfortable place to sleep using old towels, bedding, and other supplies in the pad or at the thrift store.
Don't chain your dog out; presumably you're getting a dog to have a companion. Nothing says "I love you" like being staked out exposed to the elements all day. Your pet will just become territorial and hostile. If you must keep Fido outside make sure he's got shelter, a soft dry place to lay down and some place to take care of business.
Keep in mind dogs are happy eating things you might be wary of. Dogs are scavengers, carrion eaters, and will happily eat anything. Cooked rice and raw meat will make FIDO happy, you might want to Google "BARF." Keep in mind they will happily eat anything; including things they can't or shouldn't.
Service animals are NOT considered pets by federal law and can go almost anywhere no pets are allowed including apartments, hotels, and restaurants, because they aren't pets. Service animals are trained specifically to do something useful, and in this context here, in someway help a person with a disability with something they would have trouble doing for themselves. Pulling a wheel chair, guiding the blind, alerting the deaf to noises, these are all well established uses of a service animal. The Justice Department explains it pretty clearly.
First understand that a service animal is a pig magnet. If you elect to have one (or try to fake it) you will attract all sorts of attention. Every rent a cop will stop you and ask questions. Store clerks will ask you. People shopping in the mall will remember you. If they remember one person who showed up that day, it will be you. Service animals do not help you blend in. Service animals generally don't give you freedom until its the comparative sort of freedom of having help with a disability.
Now Fido's gotta act like a proper service animal; your typical service animal has a year or more "education" and obeys in a way that would bring a tear to the eye of the most hardened marine Sargent. They've gotta be clean and well fed; Fido can be kicked for smelling. Fido can't disrupt the activity you are attending, barking, farting and noisy ball licking again will get fido kicked.
The Federal ADA sets the basic ground rules for the use of service animals. At the federal level there is no certification. Getting Fido into an establishment requires credible evidence that it is a service animal, that may be a harness, a uniform identifying it as such or a verbal assurance that it is in fact a service animal. Generally they can ask:
1. Do you have a disability? You don't need to say what your disability is. 2. Was your animal specifically trained to help you with your disability?
If you can say that with a straight face and fido acts like a robot, you're in, although if you get caught don't expect your judge and jury to be sympathetic.
The penalties in the Federal ADA are largely civil. If you get shown the door your only resource is to file suit later. Many states however up the ante by making kicking Fido a misdemeanor. Many of these states also have certification requirements, usually available to residents, and will issue you a certificate. In CA you get issued a blue dog tag. The theory if the establishment makes a "trespassing hobo call" to the pigs over Fido they'll happily explain to the establishment they are the ones getting a ticket if they keep it
Now in practice this doesn't always work; local cops generally don't enforce federal rules and many states don't have certification programs for non-residents or folks without permanent addresses. And even if they sort out all out, you'll be free to do just as soon as they do a warrant check on you and double check Fido's tags to make sure his registration and rabies are up to date. Also service animals aren't exempt from leash and popper scooper laws, do you have a bag sir?
Airplanes & Housing
While passing off fido as a service animal because he makes you feel better might not pass muster with the ADA, it does for airplanes and housing. In these two situations however an Emotional Support Animal is a valid way of getting Fido on a plane.
Airplanes are exempt from the ADA, the Air Carrier Access Act governs instead. This imposes a few minor restrictions on Fido. Size for instance, generally 80 pounds is the point at which airplanes offer to put your service animal in cargo. Expect a bulkhead seat.
Housing is not exempt under the ADA, and the Fair Housing Act provides additional rights.
Generally keeping a pet in a house is regarded as a reasonable accommodation under the ADA and the Fair Housing Act if it aids you with a disability even if it wasn't specifically trained to do so, but you might need a doctor's note.
Free service animal
Anyhow, if you really are disabled and looking for a free service animal there are a couple of options. Most service animals are provided free by charities w/ documentation of disability and a number of people have been known to train their own.
Spaying and Neutering
To avoid creating more unwanted puppies and kittens that will be euthanized, you can get your pet spayed or neutered for free or at reduced cost at the ASPCA or your local Humane Society. There are other programs that also offer spaying and neutering that you can check out here and here.
Probably the cheapest place to get your pet vaccinated is the animal control office of any city. Shop around; some cities have free rabies vaccine clinics.
Most jurisdictions will have a overly self important animal control warden that will demand proof of registration and Rabies vaccination. These tags look just round metal usually with an imprinted town, state, license # and phone #. Depending on your level of transiency and documentation you might be able to claim to be from a town that lacks a license law, print your own at a pet food store, or just register with a fake name and address. Proof of Rabies vaccination (they don't care about the other common ones and rabies is the only one that you can catch from your dog.) is something that always gets checked pretty commonly (Papers please.).
If the time ever comes to euthanize your animal due to incurable (and check with a vet to make sure) illnesses, check with vets and other pet areas for the costs. Many offer very reasonable services. You can often do this yourself in a humane manner, but it's probably worth paying a few extra bucks for a pro to put down Old Yeller in a safe, foolproof and painless way.