I've read a bit about them, but can't seem to find anywhere to reserve a ticket. Can somebody point me in the right direction? It would seem like a fun thing to do over the winter for a few weeks. What are the options? Is it affordable, or are you better off buying individual flights?

Check the FT alliance forum of your choice. There is a wealth of information including explanation of the different kinds, how & where to get them (more) cheaply, how to maximise the value from them, rule interpretations and more.

As a generalization, with all the hazards of any generalization, it can be more expensive than individual discount economy flights unless you can take full advantage of all it offers. It can be cheaper than tickets in business or first, though, since the price ratio is more like 2:1 than the 10:1 or more of regular fares. What keeps business travelers from buying them instead of trans-Pacific or whatever J/F tickets is the minimum time requirement, typically ten days out of the originating country - too long for most business trips, even with weekends tacked on at both ends.

As posted, pick an alliance first, and read the threads in that alliance's forum. (Or, read several forums first, then pick an alliance.) There's also a thread someplace on RTW options that don't fit standard alliance patterns. Then, call the RTW desk or equivalent of the airline you want to book with - probably a U.S.-based member of the alliance you picked. Alternatively, find a knowledgeable travel agent; in the context of a trip like this, his/her fee will be negligible.

You can sometimes save money by starting your trip outside the U.S., since pricing varies by country, but that adds issues of dealing with a travel agency in the country in question (the airlines themselves generally won't sell you one, for a bunch of reasons) plus getting there somehow to start the trip. I don't think you want to get involved with that, especially since the biggest savings tend to come from buying the ticket and starting in out-of-the-way countries (to someone in the U.S.) like Egypt or Sri Lanka.

abcxyz, there's a recent discussion here in TravelBuzz that you can also check out:

http://www.flyertalk.com/forum/showthread.php?t=792345