These are the exercises I do for climbing. Sorry about the cloudy video.

There are a lot of different campus moves out there. These are the ones I do:
  • Bumps: Excellent low lock-off training. Start one hand on rung 1 and the other higher, say rung 4. Then the high hand bumps up to 5 and then back to 4 for the targeted number of repetitions. (I don't like the more common version of this where you start on rung 1 and then bump one hand higher and higher. That movement gets exponentially harder as you get more tired, which is usually not a productive configuration.)
  • Drops: My favorite for recruitment training. Start off-set, for example, rungs 1 and 5. The hand on rung 5 drops to rung 1, catches it, and returns to rung 5.
  • High Shrugs: Great one-arm pull training. Start off-set, for example, rungs 1 and 5. Then, the hand on rung 1 goes to the target rung, say 7. I often bump the moving hand up after hitting a rung; for example, if I first hit 6, I'll bump it to 7 and then 8. The key here is to generate the maximum amount of the continued upward force with your low (non-moving) hand. Starting off-set (i.e., skipping the first pull from a standard Up move) focuses more of the pulling power on the high hand.
  • Ladders: Basic hand-over-hand up, maybe down. Boring.
  • Low Explosives: This trains the initial portion of a long move. Starting matched on rung 1, pull as fast as possible up to the target rung and grab it. Do not match it with the other hand. As soon as you grab the high rung, let go, drop to the ground, and restart on rung 1 for the targeted number of repetitions. Do not rest between drops and use the same side (right/left) for the entire set. (The more common of this exercise, "Touches" involves dropping back and catching rung 1. This is an ok exercise for recruitment, but too hard for other phases.)
  • Stacks: Like ladders, but much less boring. The stacks to 9 are described here and listed here.
  • Ups: The most commonly done campusboard exercise. Two hand movements, then match. Here are the standard ones used in the Lord of the Rungs competitions.

I'm convinced that easy abdominal exercises aren't worth doing. I believe you want to increase your maximum power because if your stomach muscles are contracted at over 50% of your maximum, then you can breathe. If you can't breath, you're really in trouble. (Muhammad Ali on getting tired at 0:45.) So I focus on exercises I can only do 3-12 reps of. An old German study indicated that the stomach was the second most important muscle for climbing (after the forearms.) All of these exercises train the ability to maintain tension between your feet and hands while fully extended.
  • Ab-Rollers: A great exercise only if done right. The key is to remember how you eventually want to be able to do this exercise: On your toes, body rigid, extend the arms until your face is just barely above the ground, then wheel it back in. Thus, you can't be using your hamstrings and the only motion is in the shoulders.
    • Knees: These aren't super hard, even with good form. Good form means a straight line at all times between the knees, hips, and shoulders - no movement. Also, the feet should be curled up and kept near your butt. Otherwise, people have a tendency to cheat by curling their feet up during the motion, which helps. It is critical not to bend at the hips as that can very easily lead to injury of the lower back. Once you're proficient, you can rotate slightly to the left and right for a better oblique workout.
    • Real/Stand: One problem with ab-rollers is that after working them from your knees, the next step is a lot harder. I can often do 3 sets of 25 from my knees and still not be able to do one from the stand. The other problem is that if you fail, you face plant. Fortunately, there's a solution for that: simply start from a place where the wheel will hit the wall before your face hits the ground. Work the exercise from there and slowly extend the distance.
  • Front Levers: A classic, rad exercise and probably the best on this list for training toe-to-finger tension. There are many ways to build up to doing a front lever; the "Knees Bent Reps" method below is the one that worked for me. I think the variations which involve moving the legs (in/out and so forth) are a waste of time as they don't train the proper body positioning.
    • Real/Static: I'd call this perfect form, from the legendary John Gill...
    • Knees Bent Reps: You'll need to be proficient at ab-rollers from your knees, as the motion is nearly identical, but these are harder. Start hanging on the bar with your calves bent at 90 degrees. As with ab-rollers, your shoulders, hips, and knees stay in a rigid, straight line throughout the exercise. Press with you arm and raise your body until it is just past horizontal, then lower under control. Do as many repetitions as you can. By the time I could do 6-8 reps, I could hold a real front lever.
  • Godoffes: I learned these from Jean-Paul Finne, who got them from the legendary Jackie Godoffe; hence, I call them "Godoffes." Start in an L-sit position on the ground with your palms flat and just a bit wider than your hips. Press up with your hands and hold the L-sit position. Once you're stable in that position, you can move on to opening and closing your legs. Finally, you can cross your feet underneath each-other; this requires better shoulder and stomach strength as you have to stay higher off the ground.
  • Overhead Push-ups: Start lying on the floor, face down. Extend your arms until your hands are touching. Now, contract your shoulders and stomach so that your body lifts off the ground with only your toes and palms supporting you.
    When these get easy, you can do them with one arm and on your fingertips, like Jack Lalanne...
  • Straight Leg Deadlifts: A strong lower back is essential for strengthening your core. It also helps prevent injury. See EMBB. I recommend going very light on these; I never do less than 20 reps, which ensures that I use a weight well within my ability.

Sort-of important for climbing...
  • Finger Lifts: The best place to start your mono training. These are one finger lifts using a platform with weight and a carabiner. You can train first knuckle strength (pictured) or second knuckle strength. I use an oval carabiner with athletic tape wrapped around the end for friction. This is a great base exercise for anyone wanting to improve their one-finger strength. I mainly do this during my foundation phases.
  • Hangboard: A Beastmaker 2000 is mandatory. Please mount it correctly, not like this pic, which I have seen at two different gyms now. (The slopers should hit the vertical board the hangboard is mounted to.)
    • Beastmaker Routines: These are in the Beastmaker mobile app, which I highly recommend. 
    • Encores: This is what the Beastmaker guys call repeaters while locked off. I mainly do these on my longer, endurance hangs.
    • Recruitment Hangs: One 6 second hang at maximum intensity and then rest for 2.5-3 minutes.
    • Repeaters: The Beastmaker format is 8 seconds on, 4 seconds off for 7 hangs, and 2.5 minutes between sets. Stevie Haston recommends hang times around 12 seconds for redpointing and 14 for onsighting. I use the term to describe just about any hang routine where the rest is 1/3 to 1/2 of the hang time and multiple hangs are done per set.
  • Grippers: Not very applicable to climbing, but super fun and inspiring. IronMind.
  • Power Forearms: There's a reason it's been around for 20 years. I'm working on OCRing the PDF.

To me, pulling is one of the most fun aspects of climbing. If a route doesn't require pulling, I'm just not interested.
  • Press-Throughs: I use this term for the range between where a pull-up ends (chin about bar height) and a tricep press starts (forearms parallel to the ground.) I work two ranges because the strength required varies noticeably between the top/high portion and the middle/medium portion.
  • Pullups: These are pretty straight forward and well documented elsewhere. Stevie Haston describes most of them here
    • Chin-Up, Narrow: Hands touching.
    • Chin-Up, Wide: Hands shoulder width apart.
    • Narrow Grip: Hands about 2-3 inches apart.
    • Off-Set
      • High/Low: Usually done on a campus board on two different rungs.
      • Left/Right: Hands touching and palms facing each other on a bar. I like these as they allow you to pull very low, like to your belly button.
    • One-Arm
    • Shoulder Width
    • Typewriters
    • Wide, Behind the Neck: Risky, but I still do them.
    • Wide, Front
  • Shrugs: I got these from Francois Legrand. Start one-arm hanging a bar. Rotate at the shoulder, keeping the arm as straight as possible. Move the free hand above the hand on the bar.

Rehab, Prehab, and Balancing
I know I don't do enough exercises in this category. Elijah found this great article on the USA Swimming website.
  • Arnold Press: EMBB
  • Lateral Dumbbell Raises: EMBB
  • Push-ups: Basic, solid shoulder and chest work. I rotate between three positions:
    • "Diamonds" with the hands together, index fingers and thumbs touching.
    • Second, with the hands just wider than the body and to the side, so that the hands are next to the rib cage; this emphasizes the triceps.
    • Finally, is the standard, classic push-up.
  • Scarecrows: This is what I call Exercise #2 from this article; since I don't like calling it "Full Can Straight Arm Lift."
  • Shoulder Shrugs: EMBB
  • Tubing (External Rotation): This is what I call Exercise #1 from this article.