Well this is all theory, cause I'm a crappy climber and usually lazy and out of shape too, but I'm firmly in the "climbing is the best thing for climbing" camp - provided we're talking primarily cragging. Obviously, alpine stuff with hardcore approaches or carrying big loads and you're going to want to focus on cardio and general endurance too which would usually require additional training.
But anyway, I agree climbing is the best training for climbing, but if you're really serious you would want to take a structured approach to it not just go climbing nilly willy without a plan. Out of all the "training for climbing" books I've read and ignored (see lazy and fat issue above) the best imo is "The Self Coached Climber: The Guide to Movement Training." It places the focus on proper movement and technique but still lays a good foundation for physical fitness training (climbing endurance, power, etc.) via different climbing exercises and an overall periodized training plan.