Red Mage is a very versatile class that can do a little bit of everything. A Red Mage can use some Melee Weapons, casting some White Mage spells, some Black Mage spells, and wearing some armor. However they have their own set of spells that can only be used by a Red Mage.
But here we're mostly telling of how to play a red Mage. If you take great interest in this information you can follow with us right now. We will typify and generalize what a Red Mage should and should not be doing in a normal battle.
Early on, Red Mage starts by casting debuff or two from a distance when soloing. And then it engages the enemy in melee, there is not much more to it.
In a party, you should focus on debuffs. There are some kinds of spells like Blind, Paralyze, Dia. Sometime the battles last long enough that the debuffs will wear off and you'll need to recast them. Having a Red Mage can take away the need for a Black Mage and White Mage to blow part of their time and MP on debuffs that work less and for a shorter period of times. For instance, debuffs are causing enemy to inflict less damage with Bio, lose attacking time when Paralyzed, miss altogether with Blind, and lose defense with Dia. And then you move into melee and start attacking. You can do backup healing since those debuffs are generally very cheap. Because the enemy is less dangerous and inflicts far less damage, this means less pressure on the White Mage to heal. More White Mage MP means more EXP chains, making Red Mage a valuable job. It's probably more efficient to have a Red Mage and a White Mage then to have two White Mages.
Red Mages become more popular once you get Convert and learn Refresh. Many times you will be invited to a party where they don't really want you to do anything except cast a few debuffs, and keep all their Mages refreshed. A Red Mage / White Mage can make an excellent Main Healer when looking to chain enemies, because they can refresh themselves, and Convert between battles. 
Strategic retreats are another area where the Red Mage shines. When shit hits the fan and you need to run, more often then not half your party will fall to a grisly fate before reaching the zone. And if the White Mage uses benediction, then she/he is mostly likely the first to fall. However as a Red Mage part of your job is to see it never comes to such a thing. There are several ways to help prevent this situation from ever occurring but your 1/6th the party, so if the other 5/6th manages to botch things up there is not a thing you can do to stop it. However there are plenty of ways to walk away from such a situation without a single death.
One is Bind, which is a level 11 spell that prevents and enemy from moving. It does not hold strong enemies for more then a few seconds but sometimes that is enough for your party to make their escape. Enemies can still use spells or ranged attacked. Not that great at higher levels.
Another is Gravity, which is a level 21 spell, this reduces the targets Evasion, and movement speed. As you may know, Monsters have a nasty habit of hitting you while giving chase, even though they seem to be too far away and you can never seem to hit them if the situation is reversed. Most enemies are exactly as fast as a human character and some are faster, there are very few that are slower. But gravity cuts down monster movement speed to half! This spell easily allows you and your party to escape the monsters drastically oversized melee range, and flee to safety.
Sleep is the third and most popular way. At level 25 Red Mages learn sleep and so unless someone damages the monster, which it stays asleep for a long time. Most of the monsters were totally immobilized by a high level Bard and Mage make use of sleep spells and songs so don't underestimate the power of sleep. Just let your party know not to hit it anymore.

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