Welcome teefn,
I am also fairly new here, but I will try to answer some of your questions until the veterans come along. If you haven't already, take a look at the Great Library (GL) at the Apolyton site (Civ2/Forums/Strategy I think) for lots of good reading. Some of it is pretty advanced though. ElephantU has posted a comprehensive "Tips and Tricks for New Players" in this forum. There is a big thread started by Scloopy that might help you, too.
It makes sense to devote one city to science because when you combine the Colossus, a library, Copernicus, etc they work better together. I don't think you need to specialize your other cities too much. You can use some for caravans, and use others (with barracks) for military units. After you master the basics, you could read the ELG in the GL, which describes some special-purpose cities I think.
Each of your cities is allowed to "support" a certain number of units for free. We are talking about units produced in that city (or "homed" to that city later on, using the city screen). Any more units past that number that will reduce the city's production by one shield per turn. Their picture on the city screen will have a little blue shield at their feet to let you know they are causing problems. The support number depends on your type of government, and maybe on the city size.
I am glad to hear you are interested in early conquest.

Me too. I think you should play around with the game a while and get used to all the elements before you take early conquest too seriously. But as a quick summary -
a) Grow yourself plenty of cities, maybe 10 to 30. Monarchy is OK.
b) Try to build elephants/crusaders ASAP and conquer with groups of those.
c) You can omit all city improvements [temples etc] to make a) and b) go faster.