How do I get into mountaineering as a backpacker.

I'm really interested in starting mountain climbing, and I have knowledge of wilderness camping/backpacking, but I have no sense of direction as to how to dip my toes into mountaineering. I've tried to wrap my head around all the different types of mountaineering/climbing, but I can't keep them straight, and I don't know which one is the best to ease my into mountain climbing. For example, what is mountaineering vs alpinism vs rock climbing vs scrambling. And I can't figure out technical vs non-technical. And are there terms for climbing "snowy mountains" (with glaciers and snowfields) and just "regular" rocky mountains?

I'm not looking to go climb Mt. Rainier or anything, I'm simply wondering what my best approach would be to get into mountain climbing.

Also, totally sucks, but right now I live in the midwest, but this summer I have 1-2 months free, and I plan on spending all of next summer in the Seattle area.