If you are not interested in the Everest Region, then I suggest that you go for the Annapurna Region... with fantastic views of the Annapurna ranges. It will also be cheaper as you will have to fly to Lukla for Everest Region treks. Annapurna Region treks usually start from Pokhara and ends there as well. Pokhara is a beautiful city and it is by the lake after the trek, relaxing that you really encapsulate your experiences. 3 weeks will be plenty, so you might consider other optional activities such a rafting, paragliding, kayaking, canyoning and so on... I have several friends working in this field whp can arrange for you the itinerary, but you can also do an exploratory trek by yourself. I would be better that you do your homework on where and what you want to do rather than be confused after you arrive in Nepal. The prices are very competitive and if you know what you really want then you'll have a easier time bargaining. If you have a large group, then I suggest you pre book especially during peak season. Otherwise, you can just walk from agency to agency to join a group or start a new group or privately. Just walk arounf Thamel in Kathmandu and you'll be able to get loads of trek options at various prices.
Pm me if you need any help.