Before I answer, make sure where you're looking to live is pet friendly. Some residential areas may not accept pets, while others may be open to four-legged companions AND may even have parks on property.
Your furry friend can roam happily through the new dog park at Blue Spring State Park (open from 8 a.m. to sunset), where, when you don't have your pet with you, you can also go swim in their spring (I've been. It's a blast!). The dog park at Riviera Oaks is another great destination (open sunrise to sunset), with a fenced area for smaller dogs, benches, watering stations, and a trail for walking, jogging, or running. If you need shade, there are plenty of places to get out of the sun and rest with your pet. Gemini Springs Dog Park (open from sunrise to sunset, free admission), a Volusia County hidden gem, is a great place to go to get out with your dog as well. Make sure your dog isn't too young, though—Gemini has a strict rule not allowing puppies in under 4 months old.
If you can't find a dog park, or want to roam in another Florida State Park outside those three, dogs ARE permitted in designated areas of all state parks, just be sure to bring a leash, and something to clean up after them with. Pets aren't allowed on the state park beaches or playgrounds.
If you have a service animal, though, they are welcome in all areas.