First point: AirPort is Apple's name for its IEEE 802.11b wireless networking products. Any and all products that implement the IEEE 802.11b standard can interoperate. 802.11b equipment is also called "WiFi" and the WiFi Association is very tough in testing equipment for compliance. In other words, if it says "802.11b" and/or WiFi, it'll work with any AirPort product, as well as any other 802.11b product.
Now, whether you buy a wired or wireless router depends on what you want to do. Wireless is definately cool, but it takes some work to get set up. Frankly a lot of people have learned a lot more about networking than they'd expected by putting in the time and effort to get a wireless network going.
One option is to buy a wireless router with a built-in wired switch. Linksys, D-Link, SMC, and others produce this kind of product, and they cost anywhere from $150-225. They combine the features of a "gateway router" with the wireless bridge and wired router functions to make a good all-in-one solution for sharing an internet connection. They don't necessarily support AppleTalk, so you may need to think about whether you'll need that for printing or whatever.