It's really hard to recommend something for a novice computer user like this. It SHOULD be as simple as "go to Best Buy/Circuit City/Dell/Walmart and get something cheap", but unfortunately, there are a lot of people out there looking to profit off your fear and lack of knowledge. There just isn't a safe, no-brain way to buy a computer, any more than there is to buying a car or house, except there are laws to protect house and car buyers, but not computer buyers.
A lot of these places are eager to sell you crap you don't need for obscene prices (1000% markup in some cases), AND charge you ridiculous rates to install stuff you could do yourself in 15 minutes if you had even the most basic mechanical skills. Installing a DVD drive is pretty trivial if you've ever put together an entertainment center and hooked up all the electronics to it.
The closest thing you can do to a no-brainer is find out the specs for whatever game you want to play, then look for a computer that has those specs and is cheap. You should be able to find a machine that can manhandle Civ4 for under $400.
I would also suggest avoiding Vista if at all possible. MS is pimping the crap out of it, and it's just not worth it in the opinions of anyone who has to service it or uses it in anything but the most superficial of capacities. The big gripe from me is its obscene requirements. You need a considerable amount of computer to run it, whereas if you used XP (or 2000) instead, your computer would operate the way you expect it to (as in, faster). But MS is trying hard not to give you that option, and some manufacturers are taking the bribes or whatever and not giving you the option either. Kinda like you can't get a proper-aspect ratio monitor any more, but I digress.
Let me try and summarize here.
Chances are, whatever the cheapest computer you can get new is going to be capable of running Civ4, as long as it doesn't have Vista on it. If you're stuck with Vista, make sure the machine comes with at least TWO (2) GB of RAM. Vista itself needs at least 1, and Civ4 needs 1. Also, try to avoid "upgrades", as they tend to want to charge you full price for the better hardware without discounting for what they have to remove to give that to you. Other than that, when you get your computer, try and learn a few things about it, so that next time, you'll feel better prepared.
What I essentailly did for my first "PC" was get something cheap, tear it apart and see how it worked, asked tons of questions, looked for ways to improve how it ran, and so on, ad nauseum.