If you want to jump right into the deep end of the pool:
Ruger Vaqueros for pistols. You can change the springs and hammers yourself and they'll run well.
A short-stroked Uberti 73 in your chosen caliber. In both pistols and rifles, .38 is cheaper to shoot, .45's give you more category choices (I think. But I'm sure somebody will disagree with me! )
If you don't think that you'll ever shoot Classic Cowboy, then a tuned single trigger SKB double.
Shallower end of the pool:
Uberti pistols. Tuned ones will run and should be less expensive than Vaqueros.
A 92 of some flavor or other. My good buddy Hunt Mountain Drifter has one that will almost run with the hot rod 73's.
A Stoeger double with polished chambers and lightened springs.
All this gibberish having been spouted, the advice to go to some shoots before you buy anything resembling a firearm is excellent advice. It'll give you a better idea of what you like the best and save you a boatload of money in the long term. Unless of course you're in the mood to buy, trade, sell, etc. until you settle on the set you want to compete with. It also depends, as was noted, on how competitive you want to be...