One of the most common injuries that climbers suffer from [other than perhaps shoulder injuries] that often limits or ends there climbing career is that of finger injuries. The reality of these injuries is that they can often be rehabbed quickly, and can be completely prevented if one is on top of their own soft tissue prehab. Our hands are in all reality relatively small and delicate tools, and the force they produce is not a by product of the hands themselves. Rather, most of the work that a hand can accomplish is powered by the forearm and so it is that this is often the source of much of our hand and finger pain. As the images below demonstrate, the musculature of the forearms is complex, and it is there that we must look to solve the problem of hand pain.
The big thing to understand on this one is the fact that regular soft tissue rehab on the forearms will allow for rapid recovery from training, and greatly decrease the chance of injury. If you are waiting to get on top of this kind of work until after you already hurt then it is going to take a lot more time to get better. I would suggest finding a soft tissue therapist who knows what they are doing to go to town on your forearms. If you lack that, then spend a decent amount of time working out these tight spots, and supplement that with some contrast baths for your hands and forearms.