To save money, I would use a alpine pack for backpacking trips and alpine climbing instead of a backpacking pack for both.
I just started mountaineering last year, pretty much self-taught. To start, I READ a tons of forums/articles/trip reports and WATCHED a tons of youtube videos. I would say I'm fairly informed now, although probably spent 80 hours on what you can learn in a 8 hour class.
For gear, realize mountaineering gear is expensive and be prepared to spend at least 1000, esp for overnight trips, where you have to get a 4 season tent, 20f bag, and warm pad. For gear, I would look at the RMI, IMG, or AAI websites and see what they bring or search other people's "gear lists" on backpackinglight or other forums. I'm going to do Hood in a couple weeks and plan to bring:
Midweight baselayer top, long sleeve
Soft shell jacket
Midweight insulation jacket (in pack)
Baselayer pants (in pack)
Safety kit (whistle, first aid kit)
Water (500 ml nalgene in pocket, 2L platybottle in pack)
Other tips are:
1) Research the route obsessively, can you navigate it through a semi-whiteout?
2) I personally NEVER climb under any questionable weather, sometimes even 10% chance of precipitation, as clouds=whiteout=getting lost. Always check forecast the night before and even AT the trailhead, as things can change fast and updated. Always be skeptical of mountain forecasts.
3) Practice self-arrest obsessively. Learn different crampon techniques.
4) Alpine start is 2AM. This is to avoid rock fall and the hot sun, and returning a good margin of time (6 hours) before dark.
5) Beware of bulletproof ice because you cannot self-arrest on it. Many people have died sliding down hard ice.
6) Pace yourself on the mountain. Never let your hear-rate rise too much and take breaks only every hour. You'll be much faster pacing yourself slowly and not stopping, as opposed to climbing too fast and taking many small rest breaks. Speed=safety.
Info above is not comprehensive, so be sure to read up on other things. Good luck!