Good advice from rhs - cowboy action shooters are like drug pushers ... show up and express an interest, and they'll all be urging you to "try" their particular firearms ....
Revolvers must be appropriate single actions (of course), but whether or not they have adjustable sights will simply determine your shooter category. Shotguns can be doubles (either hammerless or with external hammers, though again that may affect category) but
without automatic ejectors ... or can be a "period" repeater like the lever-action Model 1887 or pump-action Model 1897 Winchesters - though users of such shotgunes are also limited to loading two rounds at a time. Reproductions of both the Winchester '87 and '97 are now made by Norinco, available from Marstar and other suppliers. I prefer a good double with external hammers to stay the "traditional" categories .... I use a 12 ga. Rossi coach gun (and just acquired an identical one as a "backup") though I don't believe they are manufactured any longer.
Depending on where you will be shooting, you might even be able to get started with only one revolver - here in Medicine Hat, we have introduced a shooter category based on the "Working Cowboy" category of NCOWS (National Congress of Old West Shootists) - which recognizes that the average "cowboy" would traditionally have had only one revolver at most. This involves either reloading the revolver on the clock to engage the second set of pistol targets, or engaging only one set - neither of which is a scoring problem, since each category of shooter is ranked only within their own class, anyway. (Unless the event is set up to recognize a "Top Over All" shooter - but we don't do that, largely because it is pretty much meaningless, leaving Traditional, Black Powder and Duelist (traditional one-handed pistol stance) out of the running anyway ...
(By the way, NCOWS has a much more "historically based" format than SASS, if that's the way your stick floats, but is nowhere near as large as SASS.)
Some other links which may be of interest:
Western Canadian Frontier Shootist Society (WCFSS)
Alberta Frontier Shootists (AFS)
Cowboy Action Shooting - Manitoba
Rocky Mountain Rangers, No. 4 Troop (Medicine Hat group)
Ottawa Valley Marauders
National Congress of Old West Shootists (NCOWS)