been bouldering for a couple of years and taking it more seriously last 6 months. Slowly improving. My advice based on what I’ve done to improve in order of relevance at home:
1> core and back exercises, 3 or 4 times a week.
2> stretches, if youre on a cyling forum chances are you might ride a bike and as such your hips and leg flexibility will be pants.
3> fingerboards – they are the best for upping grade but as others have said not for beginners as its easy to damage pulleys/tendons. once you’ve upped your climbing to 2 or 3 times a week then consider it as an option. Ned Feehally has a beginner set of open handed repeaters that I’ve been using but listen to the pain in your fingers and back off at the slightest twinge.
pull up and press ups are limited in actually helping you boulder, they arent an answer.
look up maris lopez for gym work, most gym work is irrelevant but she has about 6 set that allegedly help.
The best advice is the best training for bouldering is more bouldering rather than anything else. I’m about to build a woodie in my garage to train. Its addictive. From bouldering ‘socially’ for a while, by training, I’ve upped my regular grade to hitting Font 6C+ and hope to have 7a soon. The one word at the core of the whole bouldering ethos has to be Gradualism, look it up.
Good luck.