Did not see one of these threads yet, so figured I'd put this here for people to share their findings.
After early access weekend, my AST is currently sitting at level 51, with an iLvl of 116 from items that I had laying around from SCH.
Here's what I'm finding so far:
-Diurnal Stance seems to be more effective for dungeon running than Nocturnal Stance.
-Potency on AST cures seems pretty weak compared to WHM/SCH cures. I'm not an FFXIV math pro, some maybe someone can confirm this for certain. I feel like if I switch into clerics things will go south pretty quickly.
-Aspected Benefic being instant is pretty cool. Make it very easy to throw it up as soon as it drops.
-For some pulls I've felt like keeping the tank alive was a challenge in either stance, and have had to spam Benefic II. Essential Dignity (long cooldown, instant cast) is a lifesaver, but it feels like you need to go to it almost too often compared to SCH with lustrate where it was more of an "oh shit" button. The AST "oh shit" moments feel like they happen more often than other healers, and the class has a harder time recovering from them.
-It's difficult to set up good card combos. Two of the cards (Ewer, Spire) aren't very helpful for dungeons/leveling due to TP/MP regen in between pulls. This may change with endgame content when fights tend to be a bit more of a marathon. Because of the RNG, I'm only using spread to hold onto Bole for when I need it with tanks.
-The best time to set up Royal Road buffs seems to be in between pulls. Due to how long the recast is on draw, you likely won't get a second card after a pull until the mobs are weakened/some are dead. In trash combat, I'm dealing out any card I get as I get it to make the most of the cooldown. I'm doing this as well in boss fights, unless the card is Bole, which I'll spread to hold for bigger tank hits.
-In general, unless the card is Bole it seems best to just use cards as you get them.
-AST dps at 51 is low. This might change with Malefic II at 54 though. This could also be because of how much time AST spends having to spam Benefic II to prevent death.
Not sure which stats to prioritize so far, as I'm not great at the math. I don't believe nocturnal shields get the same crit bonus as adlo. Normally I'd assume that crit would be better than spellspeed due to Aspected Benefic being instant cast, but with the spellspeed changes to HoT potency and with Diurnal Stance seeming more powerful in general than Nocturnal, spellspeed may be the preferred stat.
That's what I've go so far. What is everyone else seeing? As we get more information I'll do my best to update the OP to be less of my personal early observations, and more of what the community is discovering about the job overall.