Few basics:

- Make use of Essential Dignity - it's one of the only oGCD heals healers can get until later levels, and it's a very good single-target heal. Use this before using Benefic/Benefic II.
- Diurnal Sect is generally your preferred option, as it turns Aspected Benefic into a regen, like WHM (and Aspected Helios into Medica II). You'll get Nocturnal Sect at 50, which you should switch to in 8-person content if you're paired with a WHM or another AST in Diurnal Sect (Noct brings shields, but they overwrite other shields already applied); Diurnal is generally preferred otherwise.
- Card: Bole (green tree), use on tanks; Arrow (wavy light blue icon) on jobs that benefit from extra speed (BLM loves these); Spear (lightning blue snowflake) on anyone, but especially BRDs; Balance (orange sun) on the DPS unless they're doing horribly, then on whoever's doing the most damage; Spire (yellow lightning bolt), Royal Road for the AoE card version unless people are running low on TP for some reason (AoE spam from the DPS/Tank?); Ewer, Royal Road as well unless you need the MP (and you might).
- ASTs tend to have a harder time managing MP than the other healers, so use Lucid when your MP starts to dip. Lightspeed can also help with this - it not only reduces cast time, but also halves MP costs; pairs well with healing emergencies or (once you're in the 50s) for Gravity spam against trash mobs.

At later levels, you'll get access to some nice oGCD abilities (Earthly Star, Celestial Opposition, etc.), but that should cover the more immediate needs. Most of playing AST is learning the cards and managing them, which takes practice; just make sure you're pulling a new one as soon as it becomes available.