What muscles should be sore after climbing?
It depends on if you are a new climber or someone who already has some experience in climbing.
If you are a new climber, then you will probably be more sore than people who have climbed even a few more times than you. One of the main reasons is because some of the muscles that are sore are actually muscles that most people do not use very often.
Climbing puts your body in positions it normally does not go into in day to day life, and not even in many other sports. It also requires you to use muscles differently to be able to stabilize yourself on the wall and climb your way up.
It is very normally for muscles you have never felt before to feel sore, this includes your forearms, back, shoulders, biceps, legs, and many other smaller muscles. If you are a beginner, you will probably also feel some soreness in some very small muscles that do not get used very often, they are used in climbing!
Once you get past the first few initial climbing sessions, the amount of muscles that become sore lessens, but, this really depends on your climbing intensity. If you always have very intense climbing sessions, you can expect your muscles to be pretty sore. But, as you get used to climbing, the soreness goes away, almost to the point where you are not even sore.
More experienced climbers are able to climb day after day, even, depending on the intensity at which they are climbing. This is because their muscles have gotten used to the tension climbing puts on them, and they have strengthened to the point where the normal climbing sessions do not bother them that much. They would need to have a really intense climbing session for those muscles to get sore.