That's the point of playing this game, right? Make your child be the best at what she is! Classes are courses you pay money for to get better stats/skills. They're not very stressful, and usually raise stats and skills very fast. In fact, quite a few skills can only be raised through classes.
Classes cost a lot of money though. The money mentioned for each class is the amount you spend per day.
This means that early on, you will usually go bankrupt if you try to send Olive to these. The fees raise as you spend more time taking them, making it clear that you're supposed to use them for building your stats/skills only after you establish a stable source of income or have a lot of money.Classes have four levels, with each costing more money. 6 sessions of a class raise it to Adept level. 5 sessions on this level raise it to Expert, and 5 further sessions on this finally make it reach Master. Each level usually ends up raising its relevant stats/skills faster. There are exceptions to this, like Dancing. This causes a weird situation to develop, where some classes just stop being useful beyond a point as you max out their relevant skills and there's no conventional way to make them higher.
However, at the higher levels, you can increase certain stats and skills faster, which is useful if you happen to lose a bit of a skill for whatever reason. The only good reason to keep going with certain things (Such as Magic or Dance) is to get special items relevant to those arts or to get a certain ending. With Magic you get a Power ring from Fay that gives unlimited MP, with enough Dance classes and a sufficiently high Art skill you get the Dancer ending.
If you attend any class at the age of 14, Olive gains a rival based on her highest reputation. If two or more reputations are equal, the rival is chosen based on which reputation is listed first in the status screen. Eg: when Olive's Fighter and Magical Reputations are both 320 and she attends a class at 14 years of age, she'll get Anita who specializes in fighting and weaponry, and not Wendy who specializes in magic.
Keep in mind that classes have one quirk: if you're too stressed out or can't pay attention in class on a day for whatever reason, you will not gain any stats or skills for that day even though you will still be spending money. Jobs will always raise your stats/skills even if you're doing badly for the day. This makes jobs preferable to classes at times.