We spent 3 nights in the Stubai area and did some hiking while we were there. (It was in September). Here is a web site that talks about the Seven Summits:
Only 2 of them are doable as day trips (without extra equipment). Burgstall:
We climbed Elfers. It was a great hike with a bit of a scramble at the end. Many hikers were skipping the scramble and just hiking up to where it starts. (The views are about the same) We did the scramble and it was fun. From the top (either of the scramble or the top of the hike before the scramble) you look off the back side of the mountain. There was a great hut with amazing views partway up. (We stopped on the way down). You ride a lift part way up and many hikes lead from there. There are signs that will point you to the Elfers hut and summit. (it was a sharp right as we got off the lift), so getting lost shouldn't be an issue. The trail was easy to follow, and the elevation gain was a satisfying 2500 feet. As I said, there were other walks that left from the top of the lift to other huts, so if you wanted something with less elevation gain, you could probably find something. Even going part way up and stopping at the hut would provide a nice hike with good views. (if it is clear).
We didn't do Burgstall . We had a card from our B&B that allowed us to ride the lifts for free and included a Rodelbahn that was on one lift. So we went up that lift in the morning, took a short hike and rode the Rodelbahn all they way down. (Best one I have ridden in Europe) Then we went up the Stubai Glacier. The weather was turning, and we had not researched any hikes from there, so we ate dinner and came back to our B&B. You could look at this site for hikes:
It looks to me like a lot of the hikes leave from the base and part way up. The glacier may require more technical equipment. Overall, if I went back, I would not go back up the glacier. (full disclosure: We rode lifts up to several glaciers on this trip and have visited many more on previous vacations, so just seeing a glacier isn't new to us. ) I didn't think the views from the top were as pretty as the other lifts we rode.
Overall, we loved the Stubai area and would return. (It was very cost effective for us since out B&B provided the Stubai card for the lift. My memory is that these lifts are more reasonable than the ones in Switzerland). Here is the price page for the Elfer lift:
Enjoy your trip!