>>1046591The cheapest way is to find someone else who knows more than you. It's not exactly easy or reliable to do it like that.
You can self-teach, there are plenty of textbooks about it. Grab a friend and learn together-- it's much safer to go with 2-3 than 1. Most non-crazy mountains really just require you know some crevasse rescue techniques and general route-finding. Most can be learned in a general intro-to-mountaineering weekend class.
There was some cross-over from my rock climbing experience, but most of what I learned was from a weekend crevasse rescue course. These types of classes are great for meeting like-minded people who want to learn together, so be friendly and you might end up with a few climbing buddies.
Once you have some basic knowledge, look for the easiest technical peak in your area, study as much on it as you can, look for trip reports, get some route beta, get a few friends, and go for it on a good weather window. Keep going for more gradually more difficult / technical peaks. You'll get a lot of good experience this way.
The rest comes down to gear-- most of which can be bought used but in good condition on sites like GearTrade. Mountaineering boots are stupidly expensive, but I was able to snag a pair of La Sportiva Nepal GTXs for $300 on GT (MSRP $550), added bonus is they were already broken in.
