The Red Mage is a hybrid job that can wield mid-level White Magic and Black Magic. Its attributes range from average to decent and it can use most weapons adequately. In many senses, it's a jack of all trades, but closer to a "master" than a "jack".
Early on, Red Mage is pretty sweet since it can fulfill two crucial roles (healer and magician) at the same time. Of course, it won't ever live up to the full potential of the individual roles, but it's nice while it lasts. Later, you'll be using Red Mage to acquire game-changing abilities.
Each time the character takes damage, there's a 25% chance they will gain 1 BP. Does not trigger for poison/freeze damage.
Under normal circumstances, your characters will frequently be on the receiving end of enemy attacks. So having a passive ability that occasionally boosts your BP is very nice.
Level 1: Black/White Magic Level 1
Cure restores a small amount of HP, while Aero, Fire and Blizzard deal weak wind, fire and water damage, respectively. These spells can be cast on a group, but their power will be reduced.
White Magic like Cure scales with Mind, so you've got the potential to improve its healing power, while Black Magic is generally more powerful than the equivalent Spirit Magic.
Level 2: Black/White Magic Level 2
|Command||Esuna/Raise: 18 MP Thunder: 10 MP Aspir: 1 MP|
Esuna cures a character of all ailments and Raise revives a character from KO. Together, they wipe the floor with Holy Magic at this stage.
Meanwhile, Thunder deals weak lightning damage (that can be group-cast), offering additional coverage, and Aspir drains MP from an enemy, reducing the reliance on Ethers.
Level 3: Turn Tables
Generally, you won't be evading enemy attacks unless they use an ability that can miss, so it's useless in that respect. However you can use it in tandem with the Ninja's Utsusemi to force an enemy to miss with a physical attack.
Level 4: Black/White Magic Level 3
|Command||Cura: 20 MP Aerora/Fira/Blizzara: 25 MP|
Basically stronger versions of Cure, Aero, Fire and Blizzard. Nobody's going to say "no" to healing more HP and inflicting more damage. Sure, the MP cost is higher, but you can offset it with Aspir.
Level 5: BP Recovery
Around halfway through the game, status ailments will start to become very common, so BP Recovery is handy to have. You could use the additional BP to cure the ailment or prepare a vengeful attack. Later, you can abuse BP Recovery by purposely inflicting your party with ailments.
Level 6: Black/White Magic Level 4
|Command||Curada: 30 MP Esunaga: 36 MP Thundara: 25 MP Drain: 8 MP|
Curada heals a lot, Esunaga cures the party of ailments and Drain steals HP from an enemy. In terms of magic, this is as far as Red Mages go; to wield more powerful spells, you'll need to assign another job as a secondary. Still, it's not a bad place to finish by any means.
Level 7: Revenge
You can now give Revenge to other jobs, but by this stage it's frankly too costly and ineffective. If you want to swim in BP, Turn Tables and BP Recovery are far more reliable and abusable.
Level 8: Revival
If you like living on the edge, this isn't a semi-reliable way of potentially turning the tables. Still, you generally want to avoid falling below 20% HP in the first place, so its uses are fairly limited.
Level 9: MP Free in a Pinch
Like Revival, this is a fairly dangerous ability, moreso because you need to stay below 20% HP to use it. Although there is a sneaky way to use it safely and that's by becoming a Ghost. After all, ghosts have 0 HP, which satisfies the criteria.
Level 10: In the Red
When you're Braving the maximum amount of times to slay enemy mobs, this is a killer ability to have. Although not necessarily the best. At the least, it combines well with the Hawkeye's Barrage.
Level 11: Chainspell
"Spells cast will trigger twice". That's all you need to know. The cost is steep, sure, but the results are spectacular and that's all that matters.