Weapons can be kind of tough to choose from and it can come down to which version you are playing. What I will say is always good practice with any weapon proficiency is to pick a weapon for a character and stick with it. There is no real advantage to teaching all of the characters multiple weapons.

Weapons I would focus on would be Bare-Handed, Swords, Axes, and Bows. Bare-handed is just OP once you get some real levels for it and it tends to bypass a lot of annoying defenses enemies have. Axes and Bows have pretty decent mid-to-end game tier stuff that will help you out later. Swords are always a good weapon and have a pretty decent scaling, but they really begin to shine by endgame as many of them come with really good stat boosts for your character. Not to mention the Blood Sword makes the final stretch of bosses in the game manageable.

If you plan on using a dedicated mage, I would stick to either bare-handed for their weapon, or use a staff if you insist on a weapon. Staff weapons are pretty terrible, but they don't drastically lower your mages magic accuracy like most weapons do. They only do a 5% penalty.

Since I imagine you're playing the original Famicom version, Shields are also a pretty decent thing to keep around if the game works like FFIII did. The extra defense always comes in handy.