I read that you get skill points for leveling up your party members, but I can't for the life of me figure out how to spend them. Where exactly do you do that?

Skill points or proficiency points? If you mean the former (as in the abilities your pets have), then that doesnt happen until you start exploring the Lost Grove. If you mean the latter, though, then you need to attune each of your pets to an element first (i.e. Air, Earth, Water, Fire, Light, or Shadow). From there, you can distribute them accordingly based on what strengths the particular element you chose for the pet in question provides. This is a good guide on how to do that: https://www.sylestia.com/forums/?thread=45988&page=1.

and if you mean Level up points ("you get four every level up"), then you were reading outdated information because those ceased to exist when the proficiency system was overhauled into its current form about two years ago.

I meant the latter (level up points). Thanks!