Zerker: The above answers focus on the early games in the series. Here are the plot connections for the later games (6 and beyond):
Ultima 6 is basically stand alone. However, later games (notably 7 and 9) spoil the main plot of this, if you've managed to avoid spoiling it already.
Ultima World of Adventure Savage Empire and Martian Dreams are mostly unconnected to the other games, but Martian Dreams takes place after Savage Empire and has some minor connections to the former.
Ultima Underworld is stand-alone.
Ultima VII: The Black Gate does not connect to anything prior, but introduces the Guardian, who is the the common villain for the rest of the series.
Ultima Underworld 2 takes place after Ultima VII and includes some common characters in the castle. It contains some minor spoilers for the U7 events in dialog with those characters.
Ultima VII: Part 2 takes place after Ultima VII and attempts to directly continue the plot.
Ultima VIII: Pagan takes place DIRECTLY after U7:P2, but aside from the ending/intro link, the game plot is mostly disconnected.
Ultima IX takes place after Pagan, but somewhat botches the link. Plot threads are a bit of a mess, but they do attempt to link to the other Guardian-era games. This also spoils a major event in U7:Part 2, so it should be played after that one.
So, to summarize:
If you haven't spoiled U6, play it before U7 or U9
Play U7 Part 2 after U7
Play U9 after U7 Part 2
Playing UW2 after U7, and Martian Dreams after Savage Empire is probably a good idea too. The rest can be played in just about any order.
Calling 6 standalone seems to be a stretch, the 'enemy' in that game is only a threat because of what you did in earlier games. There is also a brief but rather strong connection to the original trilogy.
Black Gate (specifically the Forge of Virtue expansion) is technically connected to 3, if only in setting.