If you want a unique experience with lots of big mountains nearby to climb with quasi-reasonable logistics checkout Chile, Bolivia or Peru.
Kathmandu is a flat smoggy city that's not easy to escape by road or airplane. If things work well, 1/2 day to escape to the trailhead. If they don't, a frequent occurrence, a full day. To climb anything in Nepal you'll have to pay. The trekking peaks, shortest and cheapest you'll likely be interested in, are $350 for 1-4 persons. Bigger peaks cost more. That's not counting your national park fee, "gratuity", etc. People have bypassed these in the past but if you are employed in Nepal not advisable.
Trekking through the villages close to Kathmandu would be awesome on the weekends but it won't be spectacular scenery.