Companions in Paradox unlock new skills and abilities as they level up Jobs and Races, and in rare circumstances, they may unlock unique skills and abilities once they reach a certain character level, or they may have those kinds of skills incorporated as the companions' exclusive skill and ability(Ex. Soapy's exclusive skill Crab Beam or the seven heroes' exclusive abilities 'Hero of the Seven Worlds' and 'Sex Battle Seal').
Skills are used in combat but only if the character has that skill type enabled through means like Jobs, Races, Accessories, Unique Traits and Abilities. For example, If Lime learns White Magic as a Priest and then switches to a Merchant, she keeps the skills she learned but can't use them again unless she switches to a job that can use White Magic like Priest or a race like Heal Slime or equips the Scroll of White Magic.
Majority of skills of sexual nature can not be learned by male companions (including Luka) or Unfortunate Monsters (excluding Amabie) even if all the prerequisites for learning them are met.
Among the Weapon Compatibility, there will be a list of weapons that 'increase' the damage of the skill further than what it is mentioned. Using a grimoire with dagger skills will still bring the 100% of the skill's power, but using one of the weapons mentioned on the pages above, it will bring the 125%, 140%, or even the 150% of power that the skill possesses.
As it should be already clear after playing for some time, Jobs and Races have a rigid requirement on which weapon can be equipped, but a more rigid requirement on which kind of skill the character in question is able to use.
While the pages mentioned in the tabs above will provide a list of the jobs and races that can utilize that type of skill, only the ones written in bold have the possibility to unlock a new skill upon leveling that job/race up, which is to say that not every race and job which has the ability to use a certain kind of skill will be useful on expanding your arsenal like a few others. The list of races and jobs over each category don't represent the ability to use a particular equipment, but a skill; please don't misunderstand.
For all information about Sorcery, check this page.
If you want to check the odds of random skills, check this page.