Any good sport routine should have some kind of cross training involved. Climbing is a high physical sport and to continuously climb up the grades takes time and dedication. Sometimes, we get injured.
Cross Training as Injury Prevention
I would recommend that someone with no prior training or sport routine experience take some kind of training with a professional and they will guide you through your routine. They are also well trained to help you get educated on your body, and how you can build your routine to fit your needs. If you're lucky, they might even be a climber and have the practical experience to help you through a Sport Specific cross training routine.
It is important that we work our pushing muscles. Also known as antagonistc pairs, one of the pair is used in the pulling action while the other is used for pushing.
Its obvious that at any level, we climbers are constantly pulling on the holds (unless you're a slab master) and puts every climber at risk of elbow injuries due to repeated movement. This constant pull on the arms causes an imbalance within the body that will lead to injury if not taken care of. To any new body to the sport: start training antagonistic groups now. It does not guarantee that you won't get injured, but taking the precaution, won't hurt you.
Most gyms have an in house trainer as well that is highly trained and effective at boosting your climbing abilities.