When you get experience, besides leveling up and increasing the skill with the weapon you're using, you can sometimes choose an extra skill when you level up. Each extra skill has its own level, being level 3 (There is also a "Last" level, which I could only reach through savestate hacking) the best you can get, increasing the chances of that skill work. Before listing the skills, you can check the characters page and see how many skills each character can have.
Now with the skills. There are 3 types of skills based on the type of weapon you use:
Double: Allows your fighter with 2 melee weapons to attack twice (one with each weapon). Have Gregorio with this one and his attacks will be very powerful.
First: Allows your fighter to attack before the enemy. This is really good when you have your fighter attack first a short range wanzer withouth having to get hit first.
Stun: Stuns the enemy for a few turns, pretty useful if you don't bother buying flash grenades.
Duel: Allows you to aim at a specific enemy wanzer part, you'll be using this a lot to end battles quickly.
Speed: For machine guns only, gives you extra bullets to attack. Great for getting experience.
Switch: Same thing as double, but for gunners. One gun after another.
Guide: Skill used for missilers, you can aim at the enemy wanzer's part. You'll be using this even more than duel!