This page of the guide to Biomutant explains what climbing techniques are available in the game. The following information will help you reach higher shelves and platforms, which can be useful if you want to explore 100% of the visited locations.
The most basic technique associated with climbing are double jumps. You can use them from the very beginning of the game, but be aware that each double jump requires a small amount of Ki energy (which fortunately can regenerate on its own).
As you traverse the game world, you will find special climbing spots with interactive grips marked in yellow. In these types of locations you must start the climb by jumping to the first grip. From now on, all you have to do is to tilt the left analog stick towards other yellow handles, and the protagonist will automatically jump to them.
Hanging ropes and ladders, which can be found in town, forts and open spaces, can also help you reach higher areas. You can use all such objects you come across.
If all the methods described above didn't help you reach a higher platform, then check the map. Once you have the map on your screen, zoom in as close as possible and try to locate the route to your chosen destination. For example, it could be a location where a road is leading up a hill or where you can jump up to more rock shelves. Biomutant often allows you to go further than you might initially think. Additionally, you don't have to worry about dying easily after falling from a great height if you make a mistake. You definitely should play around with finding different paths and climbing locations.