Should I train every skill to level V?

Usually not. Each level takes much longer than the last, and level V often adds only a small bonus.

A good rule: get skills to level IV for the things you use, and save level V for skills that unlock something you need, such as a Tech 2 module or ship, or for skills that matter in every fit.

Spend your training time on breadth first. More options beat a tiny percentage gain early on.