I tried playing the games in the order they were made, but the first trilogy has aged quite a lot. There is a clear lack of balance and quite a lot of grinding. Play them only if you are curious.
Ultima IV, Underworld, VI and VII are god starting points, but each one has some barrier of entry.
Ultima IV is the oldest one on the list, so it is the one with the most antiquated interface and graphics. It fixed the balancing and grinding problems of the first trilogy for the most part and it is also the point when Ultima began taking its definite shape. it was where the virtues got introduced and it was where the land of Brittania got defined. Before each game has its own geography, but now the geography would say mostly the same and evolve.
Ultima VI is a good starting point because its more modern engine, interactive world (you can pick up and move objects) and it introduces the gargoyle race. The biggest problem is its interface, the game is more advanced than the interface could handle. Fortunately there is an open source port called Nuvie, it features a fullscreen interface mode similar to Ultima VII. Nuvie is still in development, but it should be good enough to play through the game (projects like this tend to get stuck in eternal beta due to perfectionism, even though they work well enough)
http://nuvie.sourceforge.net Ultima VII is considered the best game in the series, but it is also the largest of the games. It is very easy for newcomers to simply drown in all the content. I guess it depends on what one considers to be the "best". Similar to Ultima VI there is a port for this one as well, it's called Exult:
http://exult.sourceforge.net The download says unstable, but ignore that, it works well. Remember, eternal beta...
Ultima Underworld is another great entry, very modern design and surprisingly good interface (read the manual about "default mode"). the only problem is that it's not really an Ultiam game. It's a great introduction to the universe though. When I got fed up of the first trilogy I played though Underworld before going to Ultima IV and I loved it.