As far as I can tell, attending classes raises random sub-skills within the skill of the class. As mentioned, it's not documented.
As mentioned, a good starting point is the school survival skills, because they unlock a lot of useful basic skills. An easy starter is to try to max out one of the social skills, and use that as a basis for determining what archetype of schoolkid your character is (troublemaker, horsey gel (aka close to your familiar), bully, social butterfly, academic, or a mix). Something else to do early is to use befriend to build your clique from the other kids, based either on their gameplay advantages if you're a powergamer or their descriptions if you want to roleplay it, and nab a few key ones before they start joining their own cliques.
The other focus early on should be raising your study skills. As others have said, the best way to do this is to train at the Venalicium library. Reach level 10 (max) in study for each class skill and you will get a bonus that makes everything else easier.
Adventures and most random events require quite high skill levels. Unless you went crazy at character creation you're unlikely to get beyond the first step or two of most adventures - or resolve any random events well - for another 3-4 months. That said, in my view it's worth doing the first couple of steps of most adventures to get them out of the way when your skills are still low, if only to make the game a bit more interesting in the early days.
Apart from that, the game is about raising skills. Skills go up fast if the corresponding attribute is high, and slow if it is low. You start the game with some skills known to you, others need to be unlocked. Class skills will mostly go up on their own, unless you have terrible attributes in the classes you are taking.