Take a scenic stroll with your dogs through one of the largest urban parks in the world, enjoying winter views and possibly stopping at a beer garden if open.
Take a scenic walk with your dogs around the fortress walls for panoramic city views; the funicular is generally not dog-accessible but walking paths are pleasant.
Take the cable car from Innsbruck up to the Nordkette mountain range; dogs allowed on the cable car with leash and in hiking areas for scenic winter mountain walks.