That is indeed still a huge area; in fact that covers most of the Alps. I'll just give a few different possibilities to consider, hopefully that will help you to focus a bit more.
SwitzerlandMountains everywhere, with routes from benign walk ups to extreme climbs. It's by far the most expensive country in the Alps, which is one of the reasons I don't go there more often. The Infrastructure is great, with many cable cars that can quickly get you high up in the mountains, and lots of refuges.
If you have glacier experience, Switzerland offers some of the best climbing in the Alps. However, since you don't have that yet, you would have to go with more experienced friends or hire a guide. If that does not put you off, I suggest the Saas valley. Some easy 4000-ers to consider are
Allalinhorn,
Weissmies and
Lagginhorn.
GermanyThe south of the country has a fairly big area with hills and mountains, though none that exceed the 3000 m mark The glaciers are almost all gone, and apart from a very few exceptions you won't need crampons anywhere during the summer season. The
Wetterstein and neighboring
Karwendel ranges, both on the border with Austria, are my suggestion. Plenty of hiking to be had there, but you can find some nice scrambles as well. And if you're up for it, there are lots of climbing routes and some vie ferrate too, but considering your experience you're probably not ready for those just yet.
AustriaI first started mountaineering in the
Ă–tztal Alps, and together with the
Stubai Alps that's my suggestion. At the northern end there are lots of mountains, including some over 3000 m, that are technically fairly easy to reach. The south is glacier country, but there are still a lot of possibilities that skirt by the glaciers, so you can watch them but don't need to cross them. My recommendation? The
Kreuzspitze, one of the highest mountain in the Alps with a trail to the summit. Apart from some lingering snow high up on the mountain, there are no technical difficulties. What makes it challenging is that it's an almost 1600 m ascent by the normal route from Vent, the village at the end of the road. But then you get some fine summit views as reward.
FranceIf you don't mind lots of other people enjoying the mountains, the area around Chamonix is the place for you. It's where I usually go when I go to the French Alps. North of the main valley are the
Aiguilles Rouges, which are hiking and climbing territory. To the south is the heavily glaciated
Mont Blanc massif. If that's your choice, you'll need a guide.
For a bit more solitude, try the
Ecrins or
Vanoise.