We went last year and had an amazing trip. I planned and booked it all independently. It was really straightforward as there's loads of choice with accommodation that can easily be researched and booked online.
I was a bit concerned about altitude, particularly as DS was only 5 at the time. Because of that we started with 2 days in Arequipa, which is about 2,300m, to help acclimatise. We then went to colca canyon for 3 nights to see condors. Parts of the canyon are very high, we spent one night at 3,600m, but some villages on the edge are much lower.
After that we went to the sacred valley and Machu Picchu. We took our time and spent 4 nights in the sacred valley, in Pisac and Ollantaytambo. Then 2 nights at Machu Picchu- we visited the site twice.
Machu Picchu and the sacred valley are below 3,000m.
We then finished in Cusco for 3 nights. That was the highest place we stayed for any length of time, i think is about 3,600m so similar to the top of Teide. By then we'd been over 2,000m for almost 2 weeks so didn't notice the altitude.
We've been to Teide too. Part of the issue there is you probably went from sea level straight to the top in quite a short time. In Peru you can plan your trip so you work up to higher altitudes over a few days.
Hope this helps. It really is an amazing place. Happy to answer any other questions.