Support Abilities are what you might call the "Magic" of Vanguard Bandits. Easy to learn, and easy to use, Support Skills take AP to use (though generally not TOO much), and they're very useful when you have a little extra AP after an attack, or when you're not near an enemy, but want to give an ally a stat boost. It's a good idea to huddle your team together for the first round of turns, so that you can pile on a bunch of status enhancements, entering the battle with the upper hand! If the battle is looking grim, there are even some Support Abilities that heal other team members! Imagine that; healing in the middle of battle!
Support Abilities differ with each stone; each stone gets different abilities. Generally, they are elemental based (meaning, a legendary stone of Earth element learns Earth Support Abilities), but there is one exception to this; Gratia. The stone that powers the Ultragunner has one Support Ability; Holy Light.
Support Abilities are learned by having a high enough leveled stone, and by having a high enough BAS stat.
About Range: "Range" refers to how many spaces away from your position a Support Ability can reach. A range of 1 means a space up, down, left or right. A range of 2 means 2 spaces up, down, left, or right (as well as 1 space in any diagonal direction). Following this pattern, you should be able to tell what Range 3 and 4 would be.
Notes: Cooling Mist targets all allies within a "Range" of 2. Support Abilities CANNOT be used on the person casting them; only allies. Also, raising a stat with a Support Ability will not teach you new attacks, even temporarily.