First off, that’s awesome that you seem to be so dedicated, that kind of attitude will get you far! Overall fitness is essential and I think, as this improves, you will naturally begin to be able to begin to do pull-ups, hangboard work, etc.
as far as the gym goes, the standard arm-day, shoulder-day, back-day will get you started.
Core strength is also incredibly important, you may notice after a solid pull up or pull-up attempt, you abs are sore. I typically end each gym session with an upper and lower abs work out.
Eventually, when your body is ready, you will be able to work in a daily pull up and hangboard regimen though that will come in time. I believe the recommendation for training with a hangboard is the ability to do 10 pull ups and do at least V4 or something like that.
Bar hangs are a good starting point, if your gym has one of those weight-assisted pull-up/dip machines, that’s a good tool to use.
Protein intake and nutrition is important as well. About 3 months after I began training specifically for climbing, I made it a point to consume around 100 G protein daily and it absolutely made a difference quickly. I am a trim guy, 5’6 145 lbs and not big by any means as bulkiness doesn’t seem to be helpful on the wall.
Do what you can and, especially since you climb as often as you do, you will begin to feel stronger and see results in a matter of weeks.
Again, keep doing your thing and great job!