Ask your shelter if they are looking for foster care for any of their pets. They often need homes to care for a pet that's not ready for adoption due to healing from an illness/injury, behavioral problems, being over stressed in the shelter, being too young or underweight for adoption and sometimes because they just don't have room in their adoption center for all of their adoptable pets.
The shelter pays for medical care for fostered pets and sometimes food, and the foster period can last from days to months (depending on why the animal is a foster), giving you plenty of experience with the animal. Talk to the staff about what you're looking for and they may be able to help you find a foster animal that you may ultimately end up adopting. Some of their foster pets may have special needs that you're not ready to deal with, but others have no special needs and make wonderful pets even if they aren't quite ready for adoption.
Generally the shelter will be looking for a foster home that has some experience with pets, so if you're a new pet owner, this may not be an option.
At the end of the foster period, if it's not working out, you can return the animal to the shelter in good conscience knowing that you gave him a good place to live while waiting to become adoptable. And maybe you'll have learned more about what you're looking for in a pet.
It can be hard to get a good sense of an animal's true personality after a few visits at the shelter, my last dog was a fostered shelter dog recovering from an injury - at the shelter he was shy and reserved, not friendly at all and shied away from any contact (possibly due to his injury). When we saw him at the shelter, they said that they were looking for a foster for him until his injuries healed, so we decided to foster him while continuing to look for a pet. Even when we took him home he was very shy at first and stayed in his crate nearly all the time...it took a few weeks at home before he opened up to become a playful, loving pet. 3 weeks into the foster program, we told the shelter that we wanted to adopt him, they kept him in the foster program until the 6 week point so they could handle his followup medical care, then processed the adoption (with the adoption fee waived since we fostered him). A year later he needed one of the pins removed from his leg, the vet knew he was a foster, called the shelter and they arranged the procedure at a very low cost, basically we just paid for consumables and paid nothing for the vet's time.