Don't go in expecting to have tea with Miraak. He's a d*uche.
Don't sleep until the MQ is over. Really. Don't.
Read the new books. Lots of lore. Whether you do it before or after the MQ is up to you; they're not going anywhere.
Make it a habit to pick up every book you see in Apocrypha, and make sure you look for them as well. Every book in the game can spawn there, as well as the new ones.
If you can, play the Morrowind expansion Bloodmoon first. I didn't, and I'm kinda disappointed I missed out on the nostalgia some others got from this. I feel like it would have made it so much better.
Don't go on very many Dragon Hunting trips while Miraak is still alive, after you meet him.
Take the time to sit on the bench next to the guy with the Silt Strider and look over at Red Mountain the distance. You'll see why. Even I, having not played Morrowind, had my mind blown by how deep and peaceful that was.
Find the guy who does Kolbjorn Barrow early on. It's possibly the longest quest, and it's LONG. Also, make sure you reserve some money for him. You can probably look up how much somewhere. You want to keep the ball rolling on that quest so you can do other stuff while you wait.
Make sure you're aware of the final boss battle glitch and use one of the two possible ways to fix it. (Though, my fix is the easiest imo since the dragons are closest and easiest to hit when you're on Sahrotaar).
Make good use out of the Black Books.
The Ebony Warrior is stronger than his armor would suggest.
Carved Nordic Armor makes you look fat.
Stahlrim is supposedly now the second best armor, behind Daedric. Not sure about weapons, but I know they're weaker than Dragon (which are the best).
Do the blacksmith's misc objective in Raven Rock as one of the first things you do. The reward is something you'll want to have in case you're exploring and stumble on Stahlrim.
Serpentine Dragons might be the new strongest Dragon, not sure though.
Dwarven Ballista's attacks ignore armor, and WILL kill you if you underestimate it. But it reloads slow, so make use of this.
When walking up the path to Miraak's Temple, followers WILL get stuck on the steps. Just be patient, and if they stop moving (because you moved too far away), move close enough that they start running toward you. Obviously the steps having cutting issues and any non-Dragonborn NPC can get stuck on them. I lost Serana for three days that way, because she walked all the way back to Fort Dawnguard. Almost lost my wife and Garmr just yesterday that way. Had to be patient with them. After you get up the stone steps and see the wooden ramp, you're in the clear.
Don't unenchant the Champions Cudgel. It's a unique weapon, but the enchantment is on other things too. I made that mistake on my first playthrough of DB and I regret it. It looks pretty cool, for a two-hand weapon.
Take out the Miraak's Cultists quickly when they first show up in Skyrim, because you have to be in a major hold city to meet them, and they WILL kill other NPCs that get involved.
Attack the Ebony Warrior with your back to the mountain wall, not the air.

You wake up with a voice in your head and as cheap (free) labor to building a monument for the bad guy.