The flight school here at my home field offers (or did, a few years ago), a mountain flying class coupled with a check out in their DA-40s.
The class started with some classroom sessions on survival gear, survival techniques (fires, food, water), mountain weather, etc. Very useful stuff and very different from the normal private pilot syllabus. Lots of focus on some of the basic rules of mountain flying, like fly on the upwind side of a canyon to make an escape turn more possible. Stuff that you might think about on the couch, but need to know on demand in person.
Then we went flying. Generally kept it below the levels to require oxygen, letting us really feel the weather. Flew some passes in the 12-13K foot range, maybe 1000 AGL. We did get into some pretty bad turbulence and downdrafts, and it was great to experience it with an instructor for the first time. Nothing beats real hands on experience with these conditions. And the DA-40 has lousy climb performance at 13,000 feet in a downdraft. Not exactly a plane that can outclimb any conditions.
There is no teacher like experience. I would recommend seeing if you can take a class like this someplace, letting you safely "get into a little bit of trouble."