I have been living here for 6 months now and skiing almost everyday.
No need to go to Stubai Glacier, its a waste of time. The terrain is pretty shitty, nothing too steep and WAY too many old tourists. Unless you wanna ride switch on beginner runs its pointless.
Nordpark/Seegrube has a much younger feel to it. Not nearly as big as Stubai but still has a LOT of terrain. Some of the steepest in bound terrain in the alps, so you can just imagine the backcountry here. Their park is also top notch, and has one of the biggest kickers in the Alps. Nice QP and a few rails and a solid Halfpipe.
Axamer also has good terrain and a much better feel than Stubai, and its only 40 mins away from the city. Not as close as Nordpark but a nice change from Nordpark. They have a better selection of rails and jibs but their jumps are SO WEAK.
St. Anton is pretty sick I would definetly recommend it, but if you are going to venture that far from innsbruck just keep going and check out laax. its off the hook.
msg me if you have some more questions...peace