Start out like everyone else
Obviously, you're not excused from working your entire body, because if you just train certain areas, you're messing up your body, so first things first, get a regular training program, and start from there.
Rock climbing specific training
The one area where rock climbers simply excel beyond anyone else, is the ratio between grip strength and body weight. Additionally, you're going to want the ability to pull your weight upwards. This is something everyone already trains by default; pullups.
There are a lot of things you can do, and I personally like these, because of the effect is has on not only my grip strength, but my upper forearm.
Dead hangs. Find a bar where you can hang straight down without touching the floor, and challenge yourself to hang there for a total of 1 minute. If you can't do it in one go, just drop down, give yourself 5 seconds, then get up and continue. As you progress, you can increase the total time, or keep challenging yourself to 1 minute, and see if you can decrease the amount of times you fall off.
Dead holds. Same concept, but instead of hanging from a bar, you load a barbell up with some weight, and do the same thing as above. For variety, you can mix up how much weight you use and/or your goal time.
Pulling your weight
Do pullups, in all its glorious variety. There are so many ways to mix this up.
Grip orientation. You can do prone (knuckles facing you), supine (knuckles facing away), neutral (knuckles facing away from eachother), as well as some others that essentially is a middle ground between neutral and one of the others. You'll know it when you see it.
Grip width. Here, you can mix things up by varying how close your hands are to eachother; anything from touching, to shoulder width, outside shoulders, or even the extra wide. Again, you'll see the bar, and know. It often bends down at the end, so the extra wide grip is easier on your wrists.
All this is going to give you the ability to lift your weight up by an overhead grip, by training your upper back, biceps, and latissimus dorsi.
Mixing it into your workout plan
Essentially, unless you work out every single day, you can do these every time you work out. We normally work by the rule of thumb where we give our muscles 48 hours before working them again.
For instance, if you have a split where you work legs, chest, and back separately, you would work pullups as your primary exercise on back day, but on legs and chest day, you throw in pullups at the end. Not only does this ensure that you work your entire body, but it also gives you the benefit of also working pullups when your body is already exhausted. There are a lot of gains to be had when working out until you're tired, and still going at it.