I'm very glad to see someone else getting into Morrowind; It's really a great game, despite it's age.
First off, UESP Has an Article on the differences between Morrowind, Oblivion and Skyrim that you may want to take a look at. It's got some good info, and helped me out when I first started playing it.
Some of the first things you'll notice when playing is that unlike with the later games, Morrowind doesn't have map markers that point to quest objectives. Instead, NPCs usually give a lot more details on where to find things, so you'll need to pay close attention to what they say. It may also be good to ask as many questions as possible before leaving, to get more helpful information.
Morrowid also doesn't have a fast travel system, aside from Boats and Silt Striders at some of the bigger towns, so for the most part you'll have to walk everywhere. This is probably a major turn-off for some, but when you really start exploring, you'll find that Morrowind actually has a much smaller game world than it's successors (And predecessors), but rest assured that it's still packed with hundreds of quests, ruins, NPCs, and, um, Cliff Racers. LOTS of Cliff Racers.
Which brings me to the next bit: Combat is another thing that can be frustrating, especially early on. You'll probably have heard of it's infamous "hit and miss" combat system, where for every swing of a weapon you'll most likely just hit air. Yeah, not that fun...
I have also posted a Mod List over in the Classics group that you may want to take a look at. Most of the Mods on it involve making the game more enjoyable, such as improved graphics and combat.
Above all else, don't give up straight away. Morrowind might be a bit difficult at first (Especially if you're playing it after Oblivion and Skyrim), but it's still very rewarding and incredibly fun.
Mages Guild also gives you a form of fast travel.
Ald'ruhn
Vivec
Caldera
Balmora
Sadrith Mora
SAVE OFTEN... Morrowind doesn't have a great auto save feature SO SAVE YOUR GAME OFTEN!!!
Really, make sure you save often else you will come foul of this.
Three more things...
1. Internal areas do not respawn NPC's. Once you clear out a cave of NPC's then it is cleared.
2. Containers do not reset nor do areas. If you leave any item somewhere then that is where it will stay untill you decide to move it.
3. Don't sell or just dump the package which you are given at the start of the game. You will have a hard job remembering what you did with it. REMEMBER where you put it...
My game's journal lets you view entries by quest, but I'm not sure if that was an official update or part of a mod overhaul. It's a crucial feature though.
Alchemy is the most OP skill in the game without using glitches. There's no level caps or limit to the number of potions you can drink. So you can drink 100 fortify intelligence potions, then start making godly potions. Or you can show restraint and only make good potions that don't break the game
I never liked magic since it doesn't regenerate (there's a mod for that) and its hard to find/craft restore potions. I think enchanting is much better in vanilla. The charge recovers over time, casting never fails, and if you buff intelligence enough, creating enchantments rarely fails.
I second the Mage guild teleporters. Even if you're a pure warrior type, do enough quests to access them. Mark/Recall spells are handy too.
I usually find a house near some travel service and kill the owner (be careful you dont need them for a quest), then take their house to store stuff in. It's how the real estate business was always meant to be.
Ok guys Morrowind, its a great game so above all enjoy yourselves.
During the early levels, on levelling up put points into Agility, why? because its your Agility that makes your weapon or magic hit its target, tour the Guilds, join up, but be careful, some of your actions will have negative benefits for other guilds.
There is no fast travel apart from mages guild mages, and silt striders, there are spells called Mark and Recall, learn them and use them, they will save a lot of grief both early on and during later quests.
Lastly SAVE, SAVE, SAVE. if you don't you'll wish you had.
It's a very different game to Oblivion and Skyrim, but the other commenters have already posted some great tips!
Your choices at character creation are far more important than in later TES games. I've played several types of characters so if you've got a particular preference I could provide some tips. My personal favourite is up close and personal melee combat, and I've actually created a video guide that'll run you through the best possible start for a melee fighter if you want to max out your hit chance.
Yes, there's that Orc house in Balmora. She lives on the far side of the canal on the end opposite the silt strider.
She is part of a quest but its to kill her so you will actually save time eliminating her and claiming the house for yourself.
TIP: The key to the locked chest by her bed is on a ledge halfway up the stairs. Lots of storage inside and outside.
Buy two potions. 'Mark' and Recall' Drink the mark one just outside this house by the three crates. Where ever you are, drink the recall and you will teleport back here. Later, invest in the spells.
The thing about Morrowind is that you need to play with an entirely different mindset to Skyrim.
If you charge from quest to quest you'll miss most of the stuff. Spend lots of time going around talking to people. Because they have (largely) text dialogue, the important NPCs will all have much to say that will be relevant (as well as interesting). Take time and learn the politics, the culture and the various family and Great House rivalries.
There is much more depth in MW - with many more guilds and factions, some of which you can 'play off' against the others. So take your time with everything and you'll get more out of the experience
I wish I could 'unlearn' everything I know about MW and start over. The sense of discovering things is incredible, and although it's still a good game to play (especially with mods), its not quite the same when you literally know where every NPC stands and every quest object is placed.
Good luck - you'll have a blast.
Also - if you haven't done so yet - join the Elder Scrolls Classics group, read up everything and post questions when you get stuck (as you inevitably will)
Congrats you have Morrowind. A game that is over a decade old and is still a better game than anything on the market. Here are some tips. Say goodbye to family and friends you will never see them again. But seriously now.
Be very careful about who you kill. There is no such thing as an essential NPC they are all fair game and if you kill the wrong person it may very well bite you in the ass later on. Enjoy the conversation with the Dwemer he is literally the only one you will ever meet. Unlike Skyrim or Oblivion you'll find that this game will really award someone with a bit of imagination.
Save constantly, have multiple save. Get an external just for morrowind. 1TB should do.
Try to install this Morrowind Overhaul - Sounds And Graphics and play the game like a game of this generation, it is a bit tedious to install but is easy because you only need to follow the instructions, just its needed a little of time, then you will game an incredible game, even normal Morrowind is really cool, but with that Overhaul Mod its incredible. Oh and If you like to read will like this game, for me is the best roleplaying game about stats, skills and game mechanics, you have a lot of options to choose and to be better in your class you need to improve a few, better than Oblivion and Skyrim in terms of that, If you enjoy Elder Scrolls lore you will like this game and the environment is really incredible