The Nepal Mountaineering Association (NMA) has classified 33 peaks in Nepal as 'trekking peaks', a misleading name because all involve climbing. Ranging from 5650-6500m (18,537-21,325ft) some are, however, appropriate heights to combine within a trekking program. "Limited bureaucracy" peaks would be a better name.
Peak rules
To climb a mountain in Nepal you need permission. There are three different sets of peaks that are open, "Expedition peaks", and then the A and B group "trekking peaks". Any peak not on any list is closed for mountaineering.
There are 15 group A peaks and 18 group B "trekking peaks" and several hundred expedition peaks. Expedition peak rules have been refined over the years and now for peaks under 6500m no liaison officer is required. The liaison officer system is still incredibly corrupt, but at least there are better garbage policies. The royalty paid to the government of Nepal for expedition peaks now varies with altitude and the season. Note all peaks in West Nepal are free but the paperwork still results in a file an inch thick.
Equipment and safety
"One man's prudence is another man's poison." Tilman
If climbing in a trekking company group headed by a climbing guide, then harnesses and ropes will be used on every one of the peaks below, whether for crevasse danger or steep slopes. Nepali guides are accomplished at fixing ropes but only recently are beginning to be trained how to rope up properly for glacier travel or to set up a belay, or belay using a climbing rope. The only way to find out if they do know is to ask the guide to demonstrate.
If climbing without a guide, recognizing your personal limits is important, this is the Himalaya after all. Some of these peaks are 'straightforward' but what this really means is that to be safe you don't need a shop's worth of karabiners, ice-screws, snow-stakes and rock racks - just a partner, rope, a few bits of protection, experience using this gear, good weather and an overriding urge to die of senility.
The routes mentioned below cover the least difficult way up some of the main trekking peaks in the region. For proficient technical alpinists none of the routes are particularly challenging under good conditions. For safety-conscious amateurs, they have potential to provide satisfaction and experience without excessive danger.
Ask the trekking company if all staff are insured for their appropriate roles. Many cut corners here.
Acclimatization and itinerary planning
Heading straight up to high altitude for the first time, even if following the recommended guidelines, is usually a shock to everyone's system. The effort required for walking, let alone climbing, uphill at 5000m is much greater than you might think. So a sensible acclimatization program is essential: use the 300m a day rule, not only to avoid altitude sickness, but to give your body time to gain strength. Also plan in extra days for less than perfect weather, two significant factors that the majority of trekking companies ignore, and common causes of failure. Almost invariably a trek itinerary with a couple of days thrown in for a climb doesn't lead to a successful ascent. Plan the trip as a peak expedition, with the trek in and out as a means for acclimatizing and getting fit.
Climbing without support
Quite a few people want to try a peak without any sort of support, carrying all their own gear. Mostly this isn't entirely realistic. It is much better to hire at least a porter or two so that you can enjoy the trek, and this also means you don't have to be so fussy about weight. However the best reason to take a climbing sherpa or at least a porter-guide who speaks English is for route-finding. None of the trails are marked, and the routes on the mountain are not always obvious either, not even where you should place base camp, for some mountains. They will know where water is etc.
For this lightweight approach you will be best to arrive in Kathmandu and ask around local trekking agencies for one that will provide the permit and an NMA-registered guide to assist you.