A lot of people will say Morrowind right off the bat, and I agree with them that Morrowind is absolutely worth playing.
But you have to remember that Morrowind was made in 2002, and has a lot of artifacts of where gaming was at that time that may turn off new players. Personally, what I would do find some place where you can get the full versions of Morrowind and Oblivion both at a cheap price. A good place would be Steam since they're due for a spring sale sometime soon. Once you've got both the copies, make a character in both games, and then see which one you like more.
Morrowind and Oblivion are both excellent games with the same amount of depth and content as Skyrim. Both are equally worth your time, but sometimes Morrowind shows its age. Ultimately a lot of people will fall in love with both of them, so don't feel discouraged about trying Morrowind in the first place.
If you want some good information about the old games you can browse UESP which is a great wiki for all sorts of Elder Scrolls things. Just be warned that you may spoil some of the game for you if you go too deeply before experiencing everything yourself.
Yes. One of the most popular is http://www.gamesas.com/topic/1513057-relz-morrowind-overhaul-sounds-graphics-30-thread-6/
I don't use it myself, but some people like it. Here is a video, if you're interested: https://www.youtube.com/watch?v=3AiB-hz2tMM
It would help us to know on which gaming platform you are using. If using the Xbox 360/PS3 than your only option would be Oblivion. The Xbox 360 can play Morrowind which was released for the first Xbox console but it is almost impossible to find it in the bargain bins. I have been searching for it for a long time.
On the PC both games are great and are quite a bit different from each other.
I would suggest catching up a few let's play videos. Most that are new will be using mods. Quite often it is graphics mods for Morrowind and usually overhaul mods for Oblivion like OOO. But at least you get to see some game footage.
Here is a playlist for Oblivion on Xbox 360. It will be the same as a Vanilla game depending on which DLC's he uses and the ones you have.
https://www.youtube.com/watch?v=FYyV2nZW8iY&list=PLdx-MmfUterzswiJmybySBgyjMvpHFz9z
He also has a let's play of Morrowind which is using a graphics overhaul.
Sorcerer Dave currently has a lets play for Morrowind, Oblivion, and Skyrim running right now. I like his lets plays.
https://www.youtube.com/user/Jingles1215/videos
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I suggest to get the Steam Version or retail disk version of the game. If you are looking to get mods you might have issues with game distributer platforms, especially with Oblivion since you are likely going to need OBSE
Yes, MGE XE
http://www.nexusmods.com/morrowind/mods/41102/?
Also, this UESP page on the differences between the three games is a good read on what to expect
http://www.uesp.net/wiki/Differences_Between_Morrowind,_Oblivion,_and_Skyrim