The controls might seem strange at first, but basically you can grab with each hand (independently of one another) while the camera controls make you position your hands in the way you’ve directed the camera. Therefore you’ll need to use the camera together with your grabs to interact with objects in the world.
While the basic way of climbing is to slowly work your way up by moving each hand, it is possible to speed up the climbing process by jumping and quickly grabbing a higher spot on the wall before you can fall.
You can also swing while you’re climbing, and use the momentum from your swinging to propel yourself to greater heights.
If you’re having trouble climbing a particular area, look around for anything that might help you.
The secrets you can find while exploring that world can actually help you solve puzzles, so keep your eyes open for anything unusual.