Classes are a convenient way to group sets of skills and abilities
into archetypical packages. This can be useful for creating new
characters, both from the mechanical sense of picking starting
packages to build on, and in terms of deriving a character back
story.
Skills and abilities (for the most part) are not locked to specific
classes. Almost any character can learn any skill or ability.
However, the first class that provides an ability will determine
how that ability shows up in the abilities list.