The neighborhood of Bayview Park consists of two spits of land jutting out into the Banana River. One of these "isles" sits on a cul-de-sac, the other on a loop road, but both are connected to North Banana River Boulevard. The houses are upscale and each sits on the waterfront, thanks to the man made and natural canals. The beach is only a fifteen-minute walk to the east and dogs are permitted during specific times of the day.
If you need to stock the fridge, there is a Publix Supermarket one block east on North Atlantic Avenue. Here, you should be able to find most of what you need, including a pharmacy! However, if you need more than what Publix has to offer, you can head south to the West Cocoa Beach Causeway, which is home to a small shopping center anchored by a Winn-Dixie supermarket. Across the causeway in Merritt Island is the Merritt Island Mall and a Walmart Supercenter.
Bayview Park is a working family community, which means that there will be at least some traffic during the morning and evening hours. Once the few minutes of traffic dies down, you and your pup aren't likely to run into much else. On top of this, there are sidewalks on each "isle" for you and your dog to use for your walks.
You may encounter children at play during the day who want to come and pet your pooch and a short leash is always going to be your best choice when walking in any community area. Keeping your pup close to you also gives them a sense of security that should reduce the risk of misbehavior. The best time of day to go out for walks is during the early morning when it is still cool and there isn't much traffic around. The flat sidewalks make it easy to explore various residential roads of this neighborhood.
The nearest "green" space where you and your hound can go for long walks is going to be the beach. You can be in the parking lot at the Westgate Cocoa Beach Pier in less than five minutes by car. Unlike the neighbors to the north, here in Cocoa Beach, dogs are permitted during specific times of the day. Your dog must be on a leash and you must clean up after them, but the two of you can walk for miles to the south when you're able to stop by!
If you prefer an off-leash dog park, the Cocoa Beach Public Dog Park is less than ten minutes to the south. This park offers a fenced-in area that covers almost an acre and is divided to provide spaces for large and small breeds, each of which has water for the pups and lots of shade for everyone to enjoy.