Some recommendations...The Dolomites are the most beautiful mountains I've ever seen, so you're in for a great travel experience. As far as skiing goes, St. Anton might be better if you're totally hardcore, but the Dolomites have plenty to offer as a skiing and travel experience.
I've stayed, and skied, in Cortina, and skied Arabba/Marmolada in early March, and I have several Italian friends who would recommend Val Gardena. Short version: if I were going to the Dolomites with my Italian buddies, I'd go to Val Gardena, but with my wife - Cortina. We were there in March 2002, and my wife is already mad at me for not taking her back there last year.
1) Cortina is a great place to stay. Beautiful valley & people, lots of dining/shopping, and the views are absolutely stunning at sunrise/sunset. Good wine bars and cafes, and don't forget to have some of the fruit flavored grappa, too. The Hotel Menardi has a great breakfast and dinner, and is wonderful. When we went, $220/night included great room, breakfast and dinner for 2, and all taxes. The place has nice touches like additional, individual down comforters. It's near the lift that takes you to the Tofana area. All that said, you'll want a car in Cortina, but there's plenty of places to park. Some of Cortina's better restaurants are just outside the town center.
2) Cortina skiing: It's secondary to everything else. There are several separate areas that cannot be connected (Tofana, Faloria, and Cristallo), mostly groomers, but with good mountain restaurants (refugi). If the weather is sunny, then they'll be more people hanging out than skiing. Definitely not crowded, which is a welcome change from Arabba, where you constantly battle through the lines.
That being said, Forcella Rossa on Tofana is pretty fun, as well as Olympica, which goes through a split in the rocks called Tofana Schuss that is part of the women's world cup super g course. You can really get up some speed on these two runs.
Faloria is the lift right out of the main town center carpark, and is the see & be seen spot. Lots of elegant skiers, men and women, making beautiful GS turns. Cristallo seems to have the fewest runs, but does have a more expert slot down the center, but you might skip it & go elsewhere (or back to Tofana, if you want to stay around Cortina). [FYI: The top of Tofana was the meeting place in the James Bond film For Your Eyes Only]
In addition, if you go up to Passa Falzarego, you can take a long run down & hook up to a horse drawn section, which is pretty cool. From there, you can get a cab ride back or keep going. Ski Europe has a description of it.
Arabba/Marmolada/Selle Ronda: The Selle Ronda is a better concept than actuality. Although it's cool to see where WWI battles were fought, to do the Selle Ronda puts you through a day of long lines, and little vertical. IMO, not worth it. Arabba has some good steeps, but late in the day they have shadows. If you go off the backside of Arabba, and straight over to the Marmolada, it had awesome snow/views. Marmolada was the highlight of a couple of days there.
Val Gardena/Ortisei: My Italian friends all like Val Gardena as a place to ski better than Cortina. I've looked at the run/lift system, and I'd agree with them. So, if you're all about skiing, as opposed to some of the other travel experiences, then Val Gardena is nice.