Early on in the game, you learn that if you equip new weapons, you can permanently learn the skills attached to them. Since these skills can be really valuable throughout the game, it's good to learn as many as you can.
The best way to get many weapon skills is to create weapons through item synthesis. This can be done at any of the shops throughout the game. Use old weapons to create new, better ones. Once you have a stock of weapons, start learning the skills.
Characters learn weapon skills by fighting in battles. Even if you don't have a character in your battle party, but they're still in reserve, they're weapon skills will level up. To make things go a little faster, I suggest spending some time fighting battles for this exact purpose. Not only will you get those skills, but you'll also level up your characters quite a bit in the process.