Once you're ready to bring your new pet home, search around on sites like Craigslist.org for supplies such as used crates, scratching posts and other cheap pet paraphernalia. When it comes to toys, consider making your own. Sure, pet stores offer fancy-looking chew toys and pricey packets of fuzzy mice, but your pets most likely won't make much of a distinction between those and old braided-together socks or milk carton lids. If they can chew it or bat it, they're good. On a similar note, toy with the notion of growing your own catnip or baking your own dog treats.
As for pet-related chores like training, grooming, nail clipping and teeth brushing, do those tasks yourself. Commit to walking a new dog regularly so your family doesn't need to shell out for doggie-daycare or a pet walker.
Quality pet food and pet health insurance are also frequently recommended -- you pay more upfront, but the savings can show up down the line. And speaking of food, that and other ongoing necessities like kitty litter, flea and heartworm medicine can be bought in bulk or directly from suppliers online to help defray the cost.
As for vet expenses, veterinary schools often provide discounted services. And even if one isn't in your area, it's a good idea to shop around -- vet services can vary quite a bit in price from clinic to clinic, and you may be surprised at what you find just a few miles down the road.