What's your experience level with high DA ops?
And...
How well do you feel you self-study? There's excellent mountain flying material available.
Obviously the "best" way is to get a local CFI and a little help and training, but people do successfully fly into mountain airports. In CO we aren't really "backcountry" strips, we're mostly just very high paved runways.
Eagle is the easiest out of the three you listed. There's ways to get to Eagle that are low-ish but you're still likely to be surprised at lack of performance if temps are warm, and groundspeed at landing being significantly higher than what you're used to seeing, depending on where you usually fly.
Aspen has some special procedures that you should be aware of and flying right up close and personal next to a mountain ridgeline should be something you're comfortable with for ASE. Not to mention hellish ramp fees and reservations required in winter due to a full ramp of bizjets. And frankly if you can get to Eagle, Aspen is just "up valley" from there.
I haven't done enough flying around Jackson Hole to give any meaningful tips. Photos make it look like it's in a reasonably big valley, but local weather I can't speak for.
Many club CFIs and independents around here can do a mountain crash course with the basics in a day -- and many folks plan to stop on the Front Range and see the sights and meet up with a CFI to do a little work and then continue on their trips on up into the rocks after doing a little round robin of a few of the airports with the CFI.
We push the CPA course around here pretty hard but there's other ways to get up there for a few hours with an instructor.
The main thing you learn is to fly before noon unless there's an incredibly stable and light wind airmass over the Rockies, before the turbulence starts, and always set yourself up with an "out" where crossing high terrain.
This stuff is covered in most mountain flying material. I'm partial to Sparky Imeson's stuff, myself, but theres lots out there. And Sparky died in a mountain crash, so there's a built in warning there for all of us. He was a very good mountain pilot.
You didn't say which direction you're coming from, so we can't offer a route.