You need an ISP (internet service provider), but what's available depends on where you live (in the most rural areas, it may be only dial-up or nothing!). Most ppl use either cable (from your local cable TV provider), or DSL (the phone company). A few have access to fiber (FIOS) from Verizon. So you need that minimally. Depending upon the ISP, they either provide you w/ only a modem (which can only support ONE device connected to the internet) and you bring the router, or a combination modem+router (which allows many devices to share the internet connection). Sometimes that modem+router is only for wired devices, sometimes they include wireless too. But even if it’s only wired, you can add wireless later. In a few areas, you might be able to get WiMax, which is basically widely distributed wifi access (pretty rare though, more common in Europe). You might even be able to share someone else’s internet access if they’re willing to split the bandwidth and cost (e.g., a neighbor). And if you’re very unlucky, you might only have ONE option (the others aren’t available at all).
Again, it all just depends on what’s available in your area. There are advantages and disadvantages to each in terms of performance and cost. Too difficult to predict actual costs though, but generally speaking most ppl probably spend $35-$45/month depending on speed and other features (e.g., some might include internet phone (VOIP), or discounts w/ combo packages (TV+internet+phone)).