First of all, let me state that the daily quests from phase one were still available on the PTR even after the Sanctum was taken from the Wretched. The names and quest text are a little different to reflect the changed state of the Isle, but the quest objectives and rewards remain exactly the same. It should be safe to assume that this trend will continue throughout every stage of the Shattered Sun Offensive.
The first new quest I discovered for phase two was a very simple one, especially if there are multiple people working on it (which there assuredly will be on most servers). This quest, "The Battle for the Sun's Reach Armory", tasks you with killing nearby demons until the Emissary of Hate is provoked enough to emerge from a portal. When he arrives with a yell, you need to kill him and then plant a banner over his corpse. The thing that makes this quest trivial is that everyone in the area can use the banner on his corpse and still receive credit. I haven't had to kill a single demon for this quest, since there are so many other people on the PTR. Unfortunately, because of this, I can't provide any information regarding how many demons are required to summon the Emissary.
The second new quest on Sunwell Isle, "Distraction at the Dead Scar", is simultaneously really cool and really annoying. The quest is fun and interesting because it's a bombing run over the Dead Scar where you're annihilating demons with Arcane Charges from the back of a Dragonhawk. You even get to quickly fly over Brutallus, the second boss of the new raid instance. Unfortunately, due to the crowding on the PTR, it may take as many as four runs to nuke all the demons required for the quest. This is especially annoying because the flight to and from the Scar can get pretty monotonous since it's basically on the other end of the zone. Exacerbating this problem is the fact that some of the demons on your hit list require two or three blasts to kill, ensuring that you'll need to make a couple passes. Hopefully their hit points will be reduced before this goes live.
Data on Wowhead and Wowwiki seems to suggest that the next phase of the Offensive will deal with reclaiming Sun's Reach Harbor. Unfortunately, last I checked, not much progress had been made on the current phase. The PTR is currently down, but about this time yesterday, all of the quests were less than twenty percent complete. The second phase may require more daily quests to be completed, but I'm more of the opinion that the decreased progress is due to less player involvement on the PTR. It's no longer the weekend, for one thing, and there is likely a little less enthusiasm about doing dailies than there was a few days ago. Regardless, I'll be sure to bring you more information about the third phase as soon as it begins.