You have two choices: Put a rack on top of the car, or on the back of it. The trouble with the back rack is that the Civic is so low to the ground, its going to take some work to ensure the bike wheels arent scraping the ground behind you. The trouble with a top rack is expense. By the time you get the crossbars and the bike trays and everything, youre into it for $400 or so (maybe more than the Civic is worth).
So Id get a trunk mount and hope for the best. Performances XPORT Porter Trunk Rack ($60; www.performancebike.com) is pretty much as its name impliesa two-bike rack that attaches to a sedan trunk with various nylon webbing straps and cushioned contact points so you dont mar the paint. The Saris Bones 2-Bike Rack ($120; www.saris.com) is a little more elegant-looking, with a design that arches from the trunk lid to the bumper, and the capacity for (duh!) two bikes.
Alternatively, you can check in with your local tow-hitch outfit and see what it would cost to have two-inch receiver hitch bolted or welded to the Civic. Its doable, I know that, and likely will cost $150 or so. Then you can get a receiver hitch, which has the advantage of allowing the trunk to be opened and closed without taking off the bikes and the rack. Performance to the rescue here, too, with the XPORT Free Loader, a three-bike hitch rack for the budget price of $80 (its on sale now, regularly $100).
Check out this years more than 400 must-have gear items, including a comprehensive biking section, in the 2006 Buyers Guide.