I'm an alpinist/mountaineer type, but I can't recommend this enough:
> In Training for the New Alpinism, Steve House, world-class climber and Patagonia ambassador, and Scott Johnston, coach of U.S. National Champions and World Cup Nordic Skiers, translate training theory into practice to allow you to coach yourself to any mountaineering goal. Applying training practices from other endurance sports, House and Johnston demonstrate that following a carefully designed regimen is as effective for alpinism as it is for any other endurance sport and leads to better performance. They deliver detailed instruction on how to plan and execute training tailored to your individual circumstances.
> Whether you work as a banker or a mountain guide, live in the city or the country, are an ice climber, a mountaineer heading to Denali, or a veteran of 8,000-meter peaks, your understanding of how to achieve your goals grows exponentially as you work with this book. Chapters cover endurance and strength training theory and methodology, application and planning, nutrition, altitude, mental fitness, and assessing your goals and your strengths.
> Chapters are augmented with inspiring essays by world-renowned climbers, including Ueli Steck, Mark Twight, Peter Habeler, Voytek Kurtyka, and Will Gadd. Filled with photos, graphs, and illustrations.
It definitely helps, but I'd put it the other way around. Like, lifting before I started climbing made my start easier. I moved through the easy routes much faster than some starters because I already have enough strength that my weaknesses are in technique rather than physical limitations. The fact that I already have good physical strength has 100% impacted my climbing in a positive way.
That's my anecdotal information. The following, on specialized training and exercises that can be good for climbing, are drawn from what I learned from this excellent book about the philosophy of training to climb.
That being said, there are lifts that absolutely help climbing from a fitness standpoint. Things that increase upper-body endurance and grip strength, as well as core power. Pullups, OHP, deadlift, all come to mind. One of the basic ideas of climbing is also using your legs to push upward rather than your arms, so it follows logically that stuff like squats that help core, stability, and leg power are good for it.
At a certain level, specialized training should take over, but unless we're talking basic fatigue, there's no reason that weight training would impede climbing ability. It's also worth noting that climbing doesn't do much for your chest, so doing chest training is a good idea.
It's a pretty well respected book. The guys who wrote it are alpine climbers, pro athletes, and coaches who have coached olympians.
And sorry but most of this is not rocket science, it's not hard to grasp the mechanisms and get a grasp of how it all works.