There is indeed a logic to it, and I'm currently working on an edutainment game meant to focus on drilling this part in particular.
Finnish has 15 noun cases, but it's probably better thought of as 4+6+5 cases. The first 4 are pretty straightforward, except for the partitive, which is kind of a catch-all case. The middle 6 correspond to certain spatial relationships. Very roughly you can imagine these as {inside, outside} × {unmoving, moving closer, moving further}. Huone = room, huoneessa = in room, huoneen = into (=moving closer to the inside of) room. That kind of stuff. The last 5 have niche, special uses. That's how I mentally imagine them at least, there are a lot of details you only pick up by reading a lot.
The trade-off is that Finnish has virtually no prepositions, which English has a lot of, and which are similarly very confusing for beginners and even intermediate English speakers. There are a few post-positions, but even these are mostly things you can pick up by ear.
Verbs have a similar story. If you've ever learned Latin, Russian or Spanish you'll feel right at home with Finnish verbs, which pack a lot of info into the conjugation, but with the benefit of requiring fewer actual words per sentence.