There's some stuff in Naxx that seems so obvious that no one should ever have trouble with it...yet your first time or two in, you are almost guaranteed to mess it up. Here are some tips which I found helped me a lot. :)
1. Turn UP your spell effects. Many bosses in Naxxramas use AoE abilities, and it's essential to be able to see them and move out quickly. Even if you must have all other settings at the bare minimum, you really can't get away with not having your spell effects up.
2. On Heigan, face the platform and strafe. This is much, much easier than facing and running forward, and makes following others less tempting. Just remember not to strafe all the way to the wall on the end sections, as you will not be able to get back out in time.
3. Don't hug Zeliek. There seem to be an awful lot of strategies for the Four Horsemen, but this one point is important. He casts a chained holy attack--a little like chain lightning, but it can jump to an unlimited number of targets and the damage increases every time. Here's a picture:
To be fair, it's a little bit simplified--you'd need more than one person standing in the wrong place for the spell to jump to anyone standing at max range. On the other hand, standing too close to the melee puts you at risk of taking damage even if the other ranged DPS are safe, and that's rough on the healer. It is best simply to NOT stand anywhere between melee range and max casting range. (Also do not stand within melee range unless your raid leader has told you to do so--having too many people in melee with Zeliek can also cause deaths.)
4. Run behind the slimes on Frogger, not between. It's tempting to squeeze in between to slimes, but even with perfect timing, latency can still be an issue. Instead, wait for a fresh slime to spawn, and run directly behind it. You should be able to make it through running diagonally from left to right, moving with the slimes somewhat.
5. During the Grobbulus fight, when you drop your poison cloud you will raise your hand. If you just can't watch your buffs to see when you're going to drop the poison cloud, watch your character--there is a distinct animation when dropping the poison cloud, giving you enough warning to get out before it ticks and kills you. (I am going to assume that you know where to run to when you get injected--this is just a little trick to make sure you don't kill yourself.)
6. Run forward and hit your jump key right as you reach the edge to get to Thaddius's platform. If you jump to soon, or walk up to the edge and then jump, you're not going to make it. Rather than relying on a friendly mage to cast slow fall, it's worth it to put the time in to practice and get it right. Blackfathom Deeps has an excellent area for practicing your jumps in--if you've been there before you should remember the broken walkway. (I know I do--I had to be summoned across by the party's warlock because I just couldn't get across.) If you're not comfortable starting with something so challenging, I'd recommend Wailing Caverns to start with--you can always go back to BFD when you're a little more confident. Once you can make all of the jumps in BFD, Thaddius should give you no trouble at all.
7. Watch your charge and keep an eye on Thaddius. Your raid may have someone on vent calling out when the polarity is about to shift or when it has shifted, but nothing beats being completely aware of the situation yourself. Make sure you have your UI set up so that even if you are healing you are able to see when he begins to cast Polarity Shift. When he starts the cast, switch to watching your charge--as soon as it changes (if it changes) run straight forward, through his legs and out the other side. Do not rely on anyone else to time your movements on this fight--vent can lag, but that doesn't mean it's the lag that will get blamed if someone crosses the charges. ;)