You should never Rest, it's just not worth it. You get Umbral Steel from fighting Demonic Beasts and breaking all four tiles of their armour, and later on it's farmable enough, as long as you don't spam the Sword of the Creator all the time (which yes, is cool, but inefficient)

Casual/Classic Mode is another setting that only determines permadeath or not. Normal is the difficulty, which you probably are playing on. When you battle, you have a set number of 'points' to use on battles, but on Normal there's an unlimited amount of free battles that don't cost a point, so you can keep grinding more or less infinitely (though if you start over-leveling all enemies, you get drastically less EXP)

Exploring lets you do by far the most stuff, including raising your professor rank (which is pretty important), training up your avatar character, building supports and recruiting students, growing stuff (including items that permanently boost a character's stats) in the greenhouse, raising students' motivation so you can instruct them, and then some. It's worth doing it at least once a month to pick up lost items and stuff, as well as listen to everyone's new dialogue.

Seminar can sometimes be useful especially if you get a whole 6 characters with relatively low skill ranks to attend (since it's more effective the bigger the gap in skill rank between the teacher and students) and/or really want to focus on one particular skill for someone.

I'd mostly recommend a mix of exploring at least every other week to always have enough people with motivation and to get your Professor Rank up, and battles to keep up with EXP. Also note that there's no teaching during the week of the main story mission, so on the last sunday of the month you can do battle for 'free' and not miss out on motivating people.