Hi
20 plus days gives you lots of options, starting with the standard teahouse treks (no restricted permits requiring, no camping needed):
Annapurna Circuit starting at Bhulbhule on the eastern side, crossing Thorung La and heading south to join the ABC trek via Poon Hill. This combines two classics into one trek though the AC has been affected by the roads on both sides. The worst part of this is often said to be from Jomsom to Tatopani; you could take a jeep to Tatopani, trek on the road regardless or follow newly developed trails (more info on this if you wish...). Overall you'll go through a variety of landscapes and cultures, cross a high pass (5416m) and get great views of mountains on the approach to the pass and at ABC itself.
Annapurna map
Jiri/Shivalaya to EBC and fly out of Lukla will also take about 3 weeks. The first leg takes you over a few high-ish passes with lots up and down which will help with fitness and acclimatisation on the upper parts of the trek. Then you can trek directly to EBC (note the best views are from Kala Patar rather than EBC itself) and back or there are a variety of route options depending on time available and will - over one or more of the high passes or at least a side trip to Chhukung.
Everest map
Will you be trekking independently, with a guide and/or porter or with an organised group?
Hope that's a start.
scoodly