The climbing world has a lot of uncertainties about core strength. The first question being how helpful core strengthening really is. Chances are you have been on a climb where you felt a need for better core strength, but there is certainly plenty of climbing you can do without needing core strength. To some degree it depends on what climbing you are doing, but overall having some core strength can be beneficial. The areas you will potentially need core, will be on steep, overhung routes, or any move (on any wall) that requires you pull in tightly to the wall to position correctly.
If you want to be a successful all around climber and not limited to flat wall or slab, chances are developing some core strength will help you. If you climb an overhung or steep route, you will likely feel your core engaging for flagging, twist-locking, and a variety of other motions. One important thing to notice is that while these moves engage your core, knowing correct positioning and technique is crucial too. If you are helpless on overhung routes, merely having a strong core probably won’t be enough. Make sure you are moving your body right. For example, try hanging off a big hold and then just throwing your hand to the next jug. It’s bound to be hard. If you pull your body into the wall, position your feet well, and twist your hip a little, this move suddenly feels easier. There is a lot that technique can do to help overcome some physical hurdles. It cannot overcome all.
The bottom line is that having core strength is helpful and useful. Spending all your time developing rock solid abs and none of your time training technique and finger strength and all the other areas crucial to climbing, will set you up for failure. Spend some time working on core, but definitely do not neglect the other elements or get overly hung up on it. When I get to the gym to train, I incorporate some core work outs into my warm up or cool down. I have time to do all my climbing and climbing specific training, I just throw in some extra time for core. If you are on a tight schedule, reserve your climbing time for climbing, and do some core on rest days when you can fit it in. The good thing is that, for the most part, core workouts won’t spend all your other energy or make it so you cannot do your other training. It can often fit in as a nice compliment.
What core work outs to do is the difficult part. There is not a whole lot out there on climbing specific core. Also I find, from personal observation, that different core work outs work for different people. Some people I know can plank forever and feel nothing. Therefore, doing 40 seconds of planks in reps probably won’t help that much. Some people find sit ups more painful than useful. To some degree it is making sure you are doing it right, but I do also think there is some room for individual differences in work outs. Having said all that, I have been trying a few said to be climbing specific or said to be really good, and while I do not have all the answers, I have some that seem to work. I used to be one of those people completely helpless on overhung caves. I first learned some better techniques. Great, I could do a move or two. Then I started some core workouts and now I can climb in caves. Not at the same level I can flat wall climb, but it is getting much closer.
One I particularly like, though again cannot solidly prove how useful it is other than my experiences, is to traverse steep walls. I use the 45 degree wall at my gym. The idea is to do between 10 and 20 moves on well spaced hand holds and foot holds to get stretched out positions. Make sure to use jugs to insure you are successful and also because littler ones aren’t necessary for this since it isn’t about fingers. You can use smaller ones that feel like jugs to you if you are a very strong climber, I guess. Just make sure you are working core and not worrying about fingers. The reason I like this is because while it gets you used to using core, it also gets you to think about proper technique and positioning since you still need both to be successful. After you got your 10 to 20 moves, rest for a few minutes. Do this three times. Make sure to get plenty of rest, because you are not working on endurance, you are working on core. You can change the difficultly level to make room for progression by starting with your hands and feet closer together if you need to and working on getting as outstretched as you can.
Another one you can try is a body curl on a hangboard. The idea is to hold the two large hangboard jugs and slowly curl your body up so your knees touch your chest. Then continue this upward motion until your knees move through your arms. Then slowly and with control lower your legs back down. Do 6 to 12 reps of this. Then rest a few moments and then try a second set. This is good because it also is designed to help improve the muscles needed for highstepping, which we see a lot in climbing. If you, like me, think that sounds ridiculous and I cannot do that, what I did was trying to lift my legs as far as I could. For me this was just towards my chest. I cannot go through my arms. I do, however, feel like my core is sore and worked out from what I can do, so I think it is beneficial. I also have hope if I keep working, maybe I will advance at it one day.
Now sits up, have never really been for me. However, I do like sit ups on exercise balls. On the exercise ball they give a wider range of motion, which works out a bit more core. I also feel them more in my core, which helps me feel better that they are working. This is the same as a sit up except balancing on the ball with your feet on the ground.
These are just a couple that I have found helpful to start. I will post more as I discover them. Also feel free to share some that work well for you.