In case of emergency, the Castor Animal Hospital is conveniently situated in the middle of the neighborhood. While there aren't any pet stores within the community, there is a pet supply store in the Roosevelt Mall which borders Oxford Circle to the northeast.
Whether you want to window shop or explore parks, Oxford Circle has a little bit of everything, including busy roads. Roosevelt Avenue is a wide road with several grassy dividers and, while you're protected from cars with a verge, crossing the street might be difficult. Avoid this, PA-73 or Cotton Avenue, and Castor Avenue if you want a more peaceful walk.
If you're willing to venture out of the neighborhood a little, Pennypack Park to the northeast features miles of trails through a natural creekside setting away from busy roads. To the northwest is Tacony Creek Park, another creekside greenbelt with trails perfect for a long walk with your pet. Both of these are great options.