It's a highly walkable area and even a popular walking destination as it's part of Chicago's "Emerald Necklace" - a series of parks, picturesque squares, and historic boulevards that form a loop, or necklace, around its beloved city. Within Logan Square, a number of antiquated greystones - an architectural staple of the Windy City - line the sidewalks, giving you beautiful sights to see while walking your dog. If refined architecture isn't on your to-see list, then you can pop into any number of divey, local haunts that serve up authentic Mexican cuisine, just be sure to call ahead and ask about patio seating, as most patios in Chicago are dog-friendly.
Chicago is full of alleyways. It's not uncommon for restaurants, bars, and even other businesses to be located within them. While you won't find that necessarily in Logan Square, you will find that many locals utilize the alleyways as handy shortcuts when traveling on foot or by car. Keep this in mind when utilizing a phone to give you directions, as some map applications will count alleyways as navigational gems.
A very pedestrian-friendly city, Chicago boasts miles and miles of sidewalks, crosswalks, and bike lanes. Logan Square is no exception. When walking around the neighborhood with Fido, you won't have to worry about having to walk in the street.
Mozart Park has play equipment as well, but it also has an open, grassy lot. If your dog is trained off-leash, considerate, and friendly, this would be a great space to burn off some energy by throwing a few frisbees or balls around.
Julia de Burgos Park is a popular local gem. A modern playground keeps the park young and active. It's a pit stop along the Bloomingdale Trail which is an extension of the 606 interurban trail that spans a great distance of Chicago. This means the trails here are paved, well-maintained, and even divided down its middle to direct heavy foot and bike traffic safely and efficiently.
Central to Logan Square is Kosciuszko Park, a great place for dog owners and walkers alike. It's one of the larger parks in the area and offers pups enough space to roam and play (although it is not fenced in).