These tips assume you have a basic idea of how the fights themselves work. They are not complete overviews of the fights, but rather hints, tricks, reminders and ideas.
Keep a HoT on yourself and nearby healers. You're going to start taking more and more damage from your stacks. Luckily you heal for more, you just have to remember to keep yourself up. Other healers without HoTs will appreciate the help as well.
Stack your HoTs, roll your Lifeblooms, spam your Nourish. Make sure you hit Swiftmend on cooldown, and don't clip your Rejuvenation. The only time you should clip your HoTs is right before you're going to hop back into the Nightmare. Your powerful HoTs are all that Dreamwalker has to mitigate her slow health drain while you're inside.
Manage your mana like a champ. Use your innervate as early as possible, and make deals before the start to get the feral druids to give you innervates and sneak into the Resto Shaman groups to drop totems on command. As your gear gets better, it's not as daunting, but in the beginning you'll really need some help.
Throw HoTs on Valithria when you have a chance, but remember that it isn't your job. The raid, for you, takes more priority than the dragon.
Throw some HoTs on portal healers. They sometimes forget to heal themselves.
Be hyper-aware of your surroundings. Avoid crap on the floor, and watch out for wayward undead. Make sure your Shadowmeld (if you're a Night Elf) is ready, and your Barkskin as well. A heroic Blistering Zombie can destroy you in one hit, so don't be afraid to GTFO.
Pre-pull, HoT with Regrowth. It ticks for a long time and provides a nice buffer. Watch Sindragosa; as she moves towards you to land, begin HoTing with Rejuv and hitting WG on cooldown.
Watch your timers. If Unchained Magic is about to go out, stop healing for just a second. If you get it, continue to not cast. If you don't get it, go back about your business until the next time it is about to go out, and repeat.
Always HoT those about to get frozen (unless you have the debuff). Help them live through the hit. Save Barkskin for when you're about to get blocked.
Monitor which healers get debuffed. If you see two or more tank healers suddenly get the debuff, take a moment to HoT up the tank. Otherwise, stay on your job. You can and will wipe if your raid dips too low and dies because raid healers are too busy with other things.
DO NOT CAST WITH THE DEBUFF. If people die while you have the debuff, that is not your fault. It is much worse, and more likely to wipe the raid, if you start gaining stacks and exploding on their face, especially during phase three. Just trust your other healers and take a breather.