I'll begin with some general tips. Although a car could be convenient for getting to resorts and trailheads, it a liability in town where you are usually better off walking. If you rent one, you want to be sure you have options for parking wherever you are staying. Additionally, some ultra cheap rental car deals look great, until you show up to pick up the car and find out you need to pay extra for cross-border travel, insurance, tolls (yes, there are many and you should check out what is and isn't covered with your rental car and be specific).
If you decide to go to Innsbruck, you should realize that its primary advantages are variety, centrality, and culture. There is no destination ski area next to Innsbruck. You're going to spend some time traveling. If you want a destination ski resort experience, especially one that maximizes the potential for a powder dump, you're probably better of staying in the Arlberg (e.g., St. Anton, Stuben, Zurs, Lech, Warth).
Finally, "know before you go". In addition to the need for avalanche safety in the backcountry, off-piste resort skiing in Europe is largely uncontrolled and should be treated as such.
Day 1: Intro to Innsbruck
You've probably arrived after a long trip from the States. Chances are your jet lagged and beat whether you've come in via plane, train, or shuttle. Enjoy an easy day walking around old town. See the Golden Roof and check out some of the well stocked but expensive adventure sports stores like The Sportler (stores closed on Sundays). If you want some history, find your way to Schloss Ambras. If you need rental gear, find Die Borse, which rents alpine skiing and alpine touring gear. For dinner, find Gasthaus Anich and get the gröstl, a Tirolean meat, egg, and potato hash, with a weisbier hell (light) or dunkel (dark).
Day 2: Stretch the legs
Catch the J-line bus to Patscherkofel, the ski area south of Innsbruck. Your skis are your bus pass as skiers are free. It's a picturesque 30-minute trip. Bring your AT skis and skip the line for passes. Skin up the resort, which is AT friendly. Start to the west of the lower rope tow and head up. There is a steep stretch above the rope tow that will require you to traverse to the east side of the run. It can be difficult if icy. Follow the locals (you'll usually have uphill company). It's a bit over 3000 vertical feet to the top of the resort. Enjoy lunch at the Patscherkofelhaus, the old stone building east of the cablecar building. Or, if you are up for it and the route is open, skin and have an espresso, coffee, or tea at the Patscherkofel Gipfelstube on the summit first. After lunch, descend via the downhill route made famous by Franz Klammer in the 1976 Olympics. Practice your edging skills. This is a north facing resort in Austria. If it is in the afternoon and it hasn't snowed in a while, you're going to need them.
|The Patscherkofelhaus (left), Innsbruck (right), and the Inn Valley|
Catch the L1 bus to Axamer Lizum, an alpine resort southwest of Innsbruck. If it's a weekend, get it at one of the stops before the University. It's about a 45 minute trip to the ski area, which is modest in size, but good for a day of skiing. When you need a break, stop at the summit Hoadl House at the top of the Olympiabahn funicular and check out their pastry bar.
Day 4: Skate ski Seefeld
Catch the train (30 minutes) from Innsbruck to Seefeld and skate ski at the site of the 2019 Nordic World Championships. There are something like 400-km of cross country trails on the plateau on which Seefeld sits, so there's more here than you can ski ina week. Find a hutte or stube to stop at for lunch. One option is the Möserer Seestubin. After returning to Innsbruck, catch a cab to Gasthaus Planötzenhof for dinner.
|Seefeld area ski trails|
Day 5: Run of Fame
Double down on the train travel and ride the rails to St. Anton. Make sure it's an early one. After exiting the St. Anton train station, walk 5 minutes to the Galzigbahn and ski the Run of Fame from St. Anton to Warth and back. It will take you all day. Make sure it's a sunny one. Bring your wallet and be prepared for crowds if it is high season. Nothing is cheap in the Arlberg.
|Above Zurs on the return to St. Anton|
Day 6: Nordkette
Chance are you need a recovery day. Ride the Nordkettebahn to the top of the Nordkette, the "northern chain" of mountains above Innsbruck. This involves taking the funicular from old town and then two trams. Have a coffee or beer at Seegrube. Enjoy the views. By now you're getting tired of Tirolean food, so dine at a Nepalese restaurant like the Everest Inn and drink some craft beer at Tribaun.
|Looking down towards Innsbruck from near the top of the Haflekar tram in April|
Day 7: Ski tour
Find a local guide or rent a car and make your own plans via bergfex.com and do a ski tour in the Sellraintal or the side valleys off the Wipptal. As usual, know before you go.
|Ski touring in the Alps south of Innsbruck|