Trying to make an involved subject as simple as possible, the Conservative Jewish rules for Shabbat are pretty much the same as any traditional approach to Shabbat observance: we avoid work as well as those things that are related to the 39 forms of labor proscribed by the Rabbis. You can read them here.
Of course, not everybody agrees about how exactly to apply these rules. For example, driving a car. One is, of course, encouraged to attend worship services and to enjoy Shabbat as a day of rest. The Orthodox forbid driving entirely, as they consider the ignition of the engine to be analogous to lighting a fire, which is prohibited. Conservative practice allows it only for going to services, as it is better to pray with a community than by one's self. Reform Judaism does not, to my knowledge, have any specific restraints on driving a car.
There are special prayers for the Shabbat, and it is traditional to have three meals: Friday evening dinner, lunch after services on Saturday morning (or afternoon, depending on how long the service goes), and then a third meal late Saturday afternoon. Emphasis is on study and rest.
Sometimes, when talking to my Hebrew School students, I will say to them: "What would it be like if you had a day where you didn't have to be anywhere? Where you got up, ate, did stuff, just based on what you wanted to do and the idea that you would spend the day hanging out?" As a rule, there is consensus that such a day would be rather agreeable. I then tell them that I was describing Shabbat.
One last anecdote: I had a teacher who observed Shabbat. He told us that one time, he decided to watch television on Shabbat. In order to do so without violating any precepts, before Shabbat began he turned the TV on, but turned down the sound and set the picture to black so that when he chose to watch it, he didn't have to turn it on, thus not violating the Shabbat. He said that while it was a perfectly kosher (so to speak) watching of television, it didn't feel like Shabbat.
As we know, Judaism is a spectrum, and each of us decides for ourselves where on the spectrum we feel most comfortable. Regardless of what we consider an appropriate level of observance, we would all do well to consider this last point: What is the spirit of Shabbat? How might we, as Conservative Jews, best promote that spirit, and benefit from it?
Cantor Socoloff