Might Start Over

[Niwashi]

... and I believe the level 15 job quest will also notify you that you can equip other weapons and become a new class

It's after the level 10 class quest of your starting class that you unlock the ability to add others. But reaching the Envoy quest at level 15 in the MSQ gives you access to all three of the starting cities, so there are more classes whose starting/unlocking point you can reach after that.