Skills are used to achieve things beyond the ken of a standard adventurer. Adventurers are already capable climbers, swimmers, jumpers, and combatants. (For example, Indiana Jones is just an base adventurer with skill in Archaeology, nothing else).
Raw abilities and unskilled tasks, as well as basic skills that fall under a class, are handled with ability checks.
A PC can easily learn a number of skills equal to their Intelligence modifier. Specialists have no limit to how many skills they can learn.
Skills are handled with a d6 roll, attempting to get equal or lower than the PC’s skill level. So a PC with a 2-in-6 climb rolls 1d6 and if they get 1 or 2 they succeed.
PCs other than specialists can learn skills through use during adventures, but mastery takes focused practice. PCs that have been doing something specific can declare it a skill and gain it at 1-in-6. The DM may decide a certain number of failures and successes with the skill during adventures can increase it up to 2-in-6, then 3-in-6, but any further advancement requires dedicated Haven turns to train and focus.
There are a set of basic Specialist skills in that class description but just about anything can be a skill as long as it can be summed up with a word or two. The only limits are no perception skills (such as search) and social skills are limited to fitting in to a certain social context, nothing like persuade or intimidate, but Courtly Manners is fine.