I have acquired the GOTY edition for my Xbox 360 and am sitting down to play it now. I am curious, what sort of things should I take note of to maximize my playing experience? What sort of things should I do/avoid to get the most out of my initial playthrough? I'm largely unspoiled but for some basic details and am really eager to get into the story's crunchiness.

I'm really excited about this guys.

Any feedback should be really appreciated.

Well you should avoid asking for extra knowledge on the forums, that's for sure! Hahaha. Have fun.

Keep your Stamina full as much as possible. It affects your chance to succeed at almost everything.
To start off with a fighting chance, only use weapons/armor/magic that you're skilled with. (your starting weapon should be at least 35)

Join a Guild, Faction or Great House that fits your skills. This will give you some easy quests and experience to get you going.
You can only join one House, so pick wisely. Redoran is for Combat, Hlaalu is for Stealth, Telvanni is for Magic.
The guilds/factions are pretty self-explanatory for the most part, or general enough for anyone.

Wandering around aimlessly in the wilderness can be very dangerous at low levels. You can encounter places with enemies much stronger than yourself.

Save often, and make back-up saves.

The GoTY Journal has a search function, which helps keep quests organized in your head.

Talking about new Topics opens up other Topics. These can lead to quests, or new info.

http://www.uesp.net/wiki/Differences_Between_Morrowind,_Oblivion,_and_Skyrim
http://www.uesp.net/wiki/Morrowind:Starting_Out - minor spoilers about the beginning areas

1. Don't kill someone unless the game tells you to, and even then, a peaceful solution may be the best in the long run

2. Lots of save slots, 5 to 10 of them at least, and cycle through them. If you have to reload to fix a quest glitch, this is how you can do that

3. Don't chase stats or multipliers or even grind your character that much. If you play the game and do the quests, your skills will come along as needed. Note, there is some grinding needed to gain faction rank, but other than that, skills take care of themselves.

4. Read the books, all of them. They will fill you in on the lore and why things are the way that they are.

5. Don't wear Ordinator Armor in Vivec. Ever.

6. Read and Heed

http://www.gamesas.com/topic/18-hints-for-beginners-new-and-improved-updated/

7. Fully explore conversation topics.

Yes, it's the GOTY edition. Just for clarity's sake.

If you join House Redoran, be careful.

House Redoran is big on honor, obedience, and responsibility.

If you get an order to do something, do it; ONLY that, and no more. It might take more running back and forth, but don't assume you know what your quest giver wants and do more than your orders say.

Killing people while doing your quests might be needed, but sometimes the reward is better if you don't. Discretion is appreciated.

Any recommended race?

What's being a stealthy build like.

I have played all races. The only thing you need to concern yourself with is each Race has preferred skill, and hence a preferred weapon or method of attack. So, what ever you choose, pick the Major and Minors that compliment that race* and then choose an attack method that fits those skills. Of course, you can throw all this out by level 10. By then you can grow your skills to make up for any deficiencies in the way you want to play as opposed to what your racial characteristics would prefer.

Sneaking is always good. Every character I have ever taken through the game gets real good at sneaking. Even if they are a big lumbering Orc Tank.

*It is not a good idea to pick Athletics and Acrobatics as a Major/Minor as you can end up leveling by just moving around the game world. So, before you built up your Attack and Defense skills, your meeting higher level enemies.

But then he'll miss out on the classic "I killed such and such and moved into their house"!

That's up to you and how you want play. Each Race does come with skill bonuses and special abilities.
Combat oriented chars are a bit easier than magic or stealth.

This site will help you make a custom character: http://morrowind.inventivegamer.com/tools/chargen/

But really, just get in the game and start playing. Any and all characters can work. Worrying about your build too much isn't really necessary.