@anthony: I can recommend Nepal out of all the Himalayan countries. Not only is it affordable and easy to travel there, but the locals are welcoming and friendly. It will be more than just a mountain experience. Its also unique in that you can do independent "tea house" trekking, which is the most affordable, where you won't need to worry about food and tents etc (which otherwise porters would have to carry for you). Costs vary considerably depending on the number of days, permits, camping vs staying in lodges, whether you use a porter, guide or a "porter-guide", whether you fly locally (like Kathmandu to Lukla), and of course which agency you decide to do it with. If you care to narrow some options down, I can get a quote for you if you want.
Bear in mind that the Himalayas do not readily offer solitude along the regular routes. The Annapurna and Everest areas are packed in peak season. You'll have to contend with locals, porters, yaks, goats, donkeys and other trekkers on the trail. Naturally you can get off the beaten track a little and there are less popular, quieter regions.
Normally the passes do not go over 5600m.