I have a cat and two dogs. I rescued my cat when I was on a trip in Tijuana, and both my dogs were acquired through Craigslist. Owners basically couldn't take care of them anymore.
My place doesn't charge for a pet deposit - just a monthly pet fee of $20. The last place I lived at charged a pet deposit and pet rent. Really, it varies. I know some apartments restrict certain breeds or have a weight requirement.
I lucked out with the place I currently have as it has a huge yard. I installed a doggie door and the dogs are able to access the yard to do their business.
I'd say if you're going to get a dog, get an older one. Puppies require much more time/attention and you really can't leave them unattended for long periods of time, even if they are crate trained. I'd also go with what others have said - wait until you are settled in - but in that case I'd say find a place that is pet friendly. I have used craigslist to find all the places I've lived at, and always check the 'cat' and 'dog' boxes to see the pet-friendly results.
Obviously, you want to ensure you are able to take a dog out on daily walks and whatnot. Dogs can get destructive if they don't have an outlet for their energy. I learned that with one of my dogs when I first got him....I would get home from work and find the place destroyed. I definitely recommend crate training or taking them out before you go to work/school.
As for the costs....I go to the weekend low-cost vaccination clinics at pet stores to get my pets their shots. I got my pets neutered at a low-cost clinic as well. I do, however, have information on the closest vets/clinics should an emergency arise. I get a huge bag of dog food which lasts about 3 months (this might vary on breed). Supplies/toys/etc. can be found on craigslist for cheap.
I'd say the costs are mostly in what they destroy. The start up costs of getting a pet are maybe around $250-300. However, if you adopt from a group, your pet should already have its shots etc.