Uberti makes several rifles suitable for cowboy action. Their Winchester 1866 and 1873 replicas are both good choices. The 1866 is chambered in .38 Special, .44-40 and .45 Colt. The stronger 1873 is available in .44-40, .45 Colt, .357 Magnum and .44 Magnum. These rifles are sold under the Stoeger and Cimarron brands as well as Uberti.
I have Uberti's 1860 Henry replica in .45 Colt; it's also available in .44-40. It's a great rifle, although it's a bit on the long side.
You'll probably want to handload if you get into cowboy action, both to hold cost down as well as to tailor your loads. I don't think there's a cheaper or easier cartridge to reload than .38 Special, although .45 Colt is not too bad, either. I'm told that .44-40 is a bit trickier to load, plus you can't use carbide dies with its bottleneck case. IMR Trail Boss is the powder of choice for cowboy action.