Red Mage is really simple. You have two bars: white magic and black magic. Each time you cast the black spell, your black bar fills up, and vice versa. You want to keep the bars balanced, so you alternate between pressing your black ability and your white ability. Once both bars are filled up, you then do your melee combo, which resets the bar. Rinse and repeat. Real simple. The fun part comes from Red Mages having a passive called "dual cast", which makes it so that whenever you finish casting an ability, your next ability will have no cast time. Thus, you want to first cast a spell with a short cooldown, and then instant cast a spell that would ordinarily have a long cast time. This allows gives you a lot of mobility.
I've been leveling up a Summoner, and so far it's about micromanaging your dots and your cooldowns. Try to make sure as many of your dots are up on as many enemies as possible, try to use aetherflow whenever it's off cooldown, etc.