Most folks start out brewing extract, which allows you to master the basics without having to worry about mash temps, etc. It also softens the monetary blow of "buying in" to the hobby, which tends to be a pretty gear-heavy one.
+1 for the suggestion of picking up the basic brewing kit from your local homebrew shop (LHBS). They will be able to give you the best advice for what you need to brew the beer you want, and will also be able to troubleshoot with you when it inevitably doesn't turn out as well as you hope. If you don't have a shop near you, you can order from one of the myriad online homebrew suppliers: Midwest, Austin Homebrew Supply, Northern Brewer, MoreBeer, the list goes on and on. Probably don't need a propane burner at this point though -- if you're doing 5 gal extract, you can get by with your stovetop.
As for a recipe, you will want to brew an ale, as the (all important) fermentation temperature is easier to manage for ales than lagers. Probably also want to do something relatively easy, like a pale ale. Any of the above retailers will have extract recipe kits, which will include everything you need to brew your first batch. Just browse the list and see what sounds delicious for drinking in about a month

.
There's lots of great information on this board and folks are generous with their knowledge. There are also many great books on homebrewing that are definitely worth your while, such as those by John Palmer, Charlie Papazian, and Jamil Zainasheff.
Cheers, and welcome to the hobby.