In reply to bigrob:
The Zillertal and Ötztal should both give you loads of choice. The Zillertal has the Olperer and the Hochfeiler (among many others) for your big snow/ice mountain days. There's also the magnificent Grundschartner Nordkante (north ridge); 700m of solid granite, V+ (but pegs and aidable) on one short section, mostly IV. It's brilliant! Also plenty of sport climbing lower down and some Via Ferrata.
The Ötztal has the Wildspitze (2nd highest in Austria); all glaciated, and a traverse gives you the full experience (not hard, very safe as glaciers go, but a long day out!). There's a lot of good sport climbing in the valley, and 7 VFs from easy to hard, steep and exposed.
Both valleys are easily accessed by public transport from Innsbruck. The huts have been open for day visitors and overnight guests since 25th May; the ÖAV recommends checking previously by phone to ask about Corona-related precautions/rules.