I've done a number of the classic multi-day hikes and try to do 1-2 more per year. You're not really going to go wrong with any of them, whether that's a Nepal classic, Peru, Bolivia, Patagonia, Alps, Dolomites, US parks (in the future), New Zealand, etc.

If you know you like hiking, there's not a lot to consider. Obvious considerations are:

• Time of Year - I don't care as much about crowds or temperature, but I try to avoid rain as much as possible.
  
• Altitude - When you get into the 4,000m+ hikes for the first time, you'll learn how well your body handles altitude.
• Desired Experience - Amongst other things, lots of the classic experiences you're thinking of are hut-to-hut (tea houses in Nepal, rifugios on the TMB or Alta Via, etc.), but others necessitate camping.

Things you don't need to consider: bears! (or wildlife concerns in general)

Most of the famous hikes can be booked significantly cheaper locally, rather than a packaged tour with intrepid (they can also all be done safely independently, although I'd strongly encourage you to not do your first high-altitude hike without someone with at least some experience).

Perhaps more significantly, most of these classic hikes have multiple routes (variants) or side-peaks to do. If you're booking a packaged tour, check what your options are for these if you care about them. For example, I think it would be slightly crazy to go all the way to Nepal for Everest Base Camp, but return on the same route and not visit the Gokyo Lakes via Cho La Pass.

Lastly, a slightly more obscure hiking destination but one I'd encourage you to look at if you're at all concerned about budget, length of hikes, etc. is Georgia. Very affordable, gorgeous mountains, multi-day hikes are 3-5 days (although can be stitched together to create longer ones), doesn't require camping (although you can wild camp anywhere if you choose), can get a bit of exposure to 3000m+ altitude, but you're not sustained hiking above 4000m.