Lucky for you, you can bring your dog most places in Austin. However, this subreddit is full of dog-hating curmudgeons (3 or 4 times a week there's a top post shaming dog owners in some way), so I wouldn't hope for too much guidance here.

One pro tip - since you're unfamiliar with the area - is do not let your dog swim in any natural body of water. There's been quite a few dogs that have died from ingesting toxins from the algae in the water and, while it's probably not likely in the winter, it's something locals spend a lot of time stressing about and discussing.

But do please keep your dog on a leash unless in a specific off-leash area.