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As someone who was also pretty fuckin scared of heights (and still is, depending on the situation) but now helps teach a mountaineering course:
Start off scrambling (aka "spicy hiking"). Keep it below Class 3. Learn to orienteer, you can get lost as fuck off trail.
A decent amount of mountaineering doesn't actually involve much exposure to heights. A lot of mountaineering is just about glacier travel and off-trail navigation in the alpine. You can climb, for instance, all five of the WA Volcanos without any actual class 5 rock or ice. That doesn't mean there aren't sketchy sections with big run-outs, just pointing out that "mountaineering" != "hiking + rock climbing" like a lot of people think it is. It's kinda a different set of skills, with some overlap, but it's not the same thing.
Working on balance can help a lot with a fear of heights. Fear of heights is actually fear if falling. Do a lot of balance exercise.
Of course exposure therapy helps too. Try rock climbing outdoors. The best exposure therapy is rappelling. Find some local top-rope crag and rappell until it's automatic. But never practice repelling without an instructor, at least not the first 50 or so times.
Endurance is huge for mountaineering. Weighted uphill hikes are the best way to train. Don't waste time with anything else.
