QooEr covered most of the basics. You generally want to get a feel for how much you'll heal with your major tools, and then try your best to ensure you don't overheal. Use your oGCDs (Tetra, Assize, Asylum) before your GCDs, and realize that your GCDs are your most important resource, even moreso than MP (Between Lucid, Thin Air, and Assize you should not have MP issues). As for heals in particular...

Cure: You should ideally never use this. In a pinch, it's more MP efficient than cure II so if you ARE positively tapped due to raises or just getting raised yourself, it's an option. If you reach for it though, know that something has gone horribly wrong.

Cure II: Your bread and butter GCD heal. Don't be surprised if this heals for 30% or more of a tank's HP. You'll basically DPS until the tank gets down to 70% or so, then toss them one of these. Tetra and Solace are basically the same with benefits, use Cure II only if they're not available.

Cure III: Yes it's part of the cure line. Yes it targets other people. Don't be fooled though, it's an AoE and will heal other people for less than a Cure II. This is without a doubt your most potent AoE heal, but the range is extremely small. It's really only usable when everyone stacks up for mechanics, but that's where it really shines.

Regen: Your best friend. This heals for more than any other single target heal, and for less MP to boot. You should be keeping this on the tank constantly during fights, as it can pretty much cover auto-attacks on a lot of bosses. It's also handy to throw on a DPS who took a hit if you know they shouldn't take any more damage for a while.

Medica: This is your basic AoE heal. Later on when you have other AoE heals, this still remains your best way to get the party healed quickly when they're scattered about. Assize and Rapture are the better versions, use them if possible.

Medica II: This is an AoE regen, it's your most effective way to heal up the party IF you know they won't take more damage for a while. If you do not have the luxury of time, use Medica instead.

Divine Benison: This can be useful to help someone survive a hit they otherwise wouldn't... but it's most useful as a supplemental "heal." It's essentially a 500 potency free heal every 30 seconds. Keep throwing it at the tank.

Plenary Indulgence: For the next 10 seconds (3-4 GCDs) your GCD AoE heals will heal for an additional 200 potency. This is useful for situations where you'll be spamming medica or cure III and need everyone topped off NOW.

As far as DPS goes, keep aero/dia up on single targets and use stone/glare on them when aero is up. For AoE packs, spam holy: It's not only great damage, but the stun helps the tank survive. Afflatus Misery nearly refunds the damage of healing rather than using stone on single target bosses (4 GCDs to deal 900 potency, whereas spamming Glare would be 1200 potency... but you helped the party live, and that's a good thing.) On AoE packs it's such a DPS gain that it's worthwhile to burn lilies on heals JUST to get the blood lily.

Lastly, a bit on mentality. Healing is a dual nature in this game. The mark of a good healer is one that can put out a ton of damage while also keeping the party alive. Both are important and neither should be neglected. Keeping people alive takes priority, as they likely contribute more to the party's overall DPS than you, but if you're overly cautious and refuse to DPS because you feel it'll hamper your healing, you're only holding yourself back.