Ultima I-III are fun games, but while technically there is an overriding story arc, it's not that important to the games themselves or for Ultima IV-IX. There are some references to the events in I-III, but not enough that you need to have played them to catch on.
Ultima IV is probably the best place to start as the story arc of the Avatar runs from IV to IX (though it is referenced that the Avatar was actually the player from I-III as well, just not called that).
Here are my thoughts on the games:
IV- Quest for the Avatar - Obviously the place to start. There is a NWN module on it, but from what I read, it was severely limited by the NWN engine so I wouldn't recommend it to get the 'real story'. Granted, I knew what I was doing since I had played it before, but I finished the entire game in something like 8 hours a few years ago. It's amazing how much quicker the game goes when you don't have the long load times of the C64 holding you back!
V - Warriors of Destiny - As much as I loved U7, Ultima V has always been my favorite. It's a dark tale, but expertly done. I'm on the fence whether to recommend U5: Lazarus or not. Don't get me wrong, I LOVED Lazarus, and they did an amazing job of recreating it. However, for a modern game, because they so painstakingly tried to recreate the character and leveling systems, it seems a bit simple and some newer players didn't like that. Obviously a conscious choice by the dev (and one I support), but if you haven't played the original, it might turn you off. Also, I found that it became very unbalanced towards the end of the game.
VI: The False Prophet - The start of the new breed of games where we were no longer confined to monochrone (or 4 color) pixels. However, be careful. The story is great, but it is possible to literally skip half the game. There are several plot holes, and when I first played it, I unintentionally skipped to about 2/3 through (hint don't go to the Gargoyle City until you absolutely have to!). The U6-project is apparently coming along nicely, and hopefully will be as good as Lazarus (it is from what I've seen so far). Due to the plot issues in the original, I'd almost recommend the remake, if it were already finsihed. (also not the two spin offs were pretty fun, though not important to the story arc).
VII - The Black Gate and Serpent Isle - Absolutely amazing. Runs fairly well in DosBox, but I prefer Exult, if for nothing more than the better scaling of the graphics.
VIII: Pagan - Sad to say I never played this one for more than an hour or two. I may try again, but I couldn't get into it. From what I have seen of IX, the story isn't that important to the overall story arc, so I may never get to it.
IX: Ascention - I've had it for 9 years and finally started playing it. When I first got it, my PC wouldn't run it. Then I tried it years later and found it awful. I'm trying again, just for the closure. HIGHLY recommend using the unofficial patch as well as the community dialogue mod which makes it tie in better to the previous ones. It's a struggle getting through it, though in and of itself, it's not a bad game, it's just not a traditional Ultima game. It's like reaching for what you think is a Coke and getting mediocre Iced Tea. It just doesn't taste like you are expecting.
I sincerely hope that once the U6: Project is done, an effort will be made to remake U9, maybe using project Britannia, with the so-called original script.