While you may not be able to enjoy a MLB game with your pet (only service animals are allowed within the ballpark), there are several breweries, bars, and restaurants that offer dog-friendly spaces so you can enjoy your beer and french fries while your dog begs for a just one bite.
Tamer during the daylight, Wrigleyville is great to visit if you're in the mood for sports bar fare and to sightsee the iconic sports arena. After nightfall, the streets become crowded with seekers of nightlife fun and pub-crawlers, especially after a winning home game.
A stroll up N. Seminary Avenue conveniently hits multiple parks in one straight line (Kelly and Challenger). This would be an especially ideal walk for a dog that's perhaps a little wary of heavy car traffic or strangers as it's through a quieter neighborhood, especially when compared to the surrounding blocks of Wrigley Field. This planned out promenade borders the Graceland Cemetary, which perhaps not appropriate for dogs, has beautiful, well-kept grounds and a small pond which are lovely to view.
Kelly Park is a small neighborhood park with a modest playground for the young or young at heart to enjoy. There's no designated off-leash area for dogs, but animals are welcome and between sidewalk paths and open, grassy space, there's room to walk or jog alongside your pup.
If your tail-wagging companion is the type of canine that needs leash-free exercise where they can run and play with others, head a little north of Wrigleyville to Challenger Park. A long, rectangular shape, the park itself is very unique. It also offers dog-owning visitors a designated Dog-Friendly Area that's fenced in. The dog park is not grassy, but your pet is unlikely to mind a space to roam and play with other dogs even if it is concrete.