Should I do missions first or the story?

If you need to get equipment for the students you're raising to proceed, do the missions first. If you want to get Pyroxenes, clear the story (skip if you don't need to watch it). If you want to level up, run missions or event mission sweeps. Also, for Total Assault and Decisive Battles, unless you're aiming for Gold or higher, I find that Tier 5-6 equipment for students is usually sufficient. Even if your students' levels are lower than the recommended mission level, you can cover up to a 15-level difference with skill.

Ah, since the story is basically everything in this game, if you get stuck while progressing the story, just do a few missions, level up, and then try the story again.

I believe the general approach is to do the missions first and then the story if you get stuck.