You learn skills independently of what you do in game -- they train constantly, one at a time, and the only impact you can have on them is to buy stat modifying implants. Grinding is either for in-game currency or for standings with NPC organizations.

Mining in lieu of missions at the beginning of the game is not really worth it even if you want to do industry and industry alone, since you don't need much in the way of skills to do the lowest levels of missions, after which you can start working your way up the station trading ladder if you really don't want to blow things up. You need the capital to buy the skillbooks for better ships and mining, and plinking away at whatever rocks the better miners have left behind with a mining frigate is slower than just rampaging through some low-level missions.

I highly recommend having at least one toon that is capable of combat, if only for self-defense. One account gets you three toon slots, though you can only train one at once. People who only do industry and can't contribute to PvP or PvE often don't make it very far when they try to find corporations to join.