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.
Make sure you and the other healers have evenly distributed points to run to during Bonestorm.
Do not be afraid to pop Barkskin; you cannot use it while on a spike anyway. If Marrowgar heads towards you, pop it and run.
Watch your timers. Start throwing out Regrowths on the raid 10 to 15 seconds before he starts spinning around. Though the HoT portion is weaker than Rejuvenation, it will last much longer. You can cast Rejuvenation as you are running around the room since you do not have to stand still.
Don't be afraid to stack full HoTs on someone spiked during Bonestorm. Try to do this especially if Marrowgar is spinning on top of them, and there is no Hand of Protection available.
Have Cyclone bound in Tree Form. Unless your guild has given you a priority mark, your main goal should be to cyclone mind controlled raid members who head into melee. They stand a good chance of being cleaved to death, with no other CC able to save them by granting them immunity.
After her mana barrier drops, the easiest way to avoid ghosts is to switch sides whenever she pops them out. Since you are a Druid, you are a mobile healer. A ghost will always target a person and chase them down. No matter where you are, if a ghost pops down and you immediately run to the other side of the room, you will not get hit by a ghost.
Curses and MC's are priority. Healing Damage is important, and you will spend most of your time doing that. But never ignore an MC running loose, and never ignore a curse on someone.
Macro Nature's Swiftness and Cyclone together. This could save someone's life.
LOL DON'T GET HIT BY ROCKETS LOL
HoT up the tank before he goes over and keep him HoT'd. HoT up the tank on deck. Play some catch up with the other raid members on board, but their damage is unpredictable and HoTs are often overwritten by Chain Heals and the like. If axe throwers are levelling up, though, buffer with HoTs anyway. They can get nasty (and are one of the few ways to wipe on Gunship).
Make sure you set up healer assignments for each mark. Have a plan for up to six marks. Paladins, then Priests, then Druids, then Shamans.
Make your Paladins call out when they are off the tanks (i.e. healing two marks). As a Druid, you should be able to help by adding HoTs to the current tank while still buffering Marked targets, also with at least one HoT.
If you are just starting HMs and your gear is on the lower end, manage your heals wisely. Your Paladins, especially, may need your Innervate more than you, and if a Mark goes down it's a wipe. It is unfortunate, but at this point your Innervate is likely no longer your own, unless you have a fair supply of Feral DPS and Moonkins.
If you are marked, move to melee range. This allows you to gain maximum benefit from AoE healing. If you are able, stand behind the boss and right click him so you begin to punch between heals. Even a small amount of Judgment of Light healing helps. Only do this behind the boss so as not to parry-gib your tanks.