I'd play though the campaign. It is, in effect, an extended tutorial. You'll learn a bit of the backstory, which is nice for immersive purposes, and by the time you have completed it you will have a pretty good working knowledge of the mechanics of the game.
As to which version to start out with; for many games, you would want to work your way through from the first, just to be able to follow the storyline. In the GalCiv series, this is much less important to the actual gameplay.
I personally would jump right in to TA. It's the version that you will end up in anyway, so you might as well start learning it now. The program is much more efficient at using your system's resourses, the graphics and effects are much better, and it is, IMO, the most interesting version. In DL/DA, all of the races share a generic tech tree, which makes all of them play pretty much the same. In TA, each race gets it's own unique tech tree, which gives each one of them a much different 'personality'.
DL/DA are a bit easier to learn at first, but TA is a much deeper game.
Good luck, and if you have any questions don't hesitate to ask.
