The accuracy demands for cowboy action are not high. So really we just need something that goes BANG! reliably. And pretty much anything works.
What is more important with fixed sight guns is that you pick a bullet weight you KNOW you can get. And once you pick your bullet weight that you push it to a consistent velocity using whatever powder works. This is important because you are quite likely going to have to file down the front sight blade to match the gun's POA to the POI of the gun and ammo you're using. And with this style of sight adjustment it's not easy to back up
So a continuous and reliable supply of at least the same WEIGHT of bullets is critical.
The ammo which will print on the targets closer to the aiming point of the stock front blade are going to be the heavier bullets driven slowly by light charges. And even those are likely going to hit a bit low. So something like 158's of some sort pushed at modest velocity is likely going to be the round that works best before any sight blade trimming is done. If you go with a lighter bullet you'll need to shave the front blade down more. And it will limit you to using that same lighter bullet from that point on. At least it does if you want to avoid using any "Kentucky elevation" in your sight picture.
So it's important to shoot a few different loadings and make some decisions on what you can find readily and reliably and then make some up to try it to find the power level you like to shoot. Some go with crazy light loads that risk not making it to the target while others, I'm one such, prefer stronger loads so we feel like we're actually shooting a gun. Many others are somewhere in between. There is a speed limit in the rules to protect the targets so if you opt for the stronger loads don't go above the muzzle velocities from pistol or rifle given in the rules. And with lighter bullets this becomes a very real risk.
Locally I've got Bullet Barn just about in my own backyard. So the obvious choice is using their product. A popular bullet from them because it feeds well in the rifles is their 130gn LRNFP. I'd say all but a couple of the local folks shooting the .38/.357 guns are using these for this reason. A side bonus is that with the amount of ammo used in matches and practices it also saves $12 or so per thousand which can go towards powder or primers.
Being more specific I load .38Spl's with this bullet and 3.8gns of Tightgroup for my pistols and the same bullet in .357Mag casings for my Rossi 92 rifle with 4.0gns of Tightgroup. The added .2gns being to make up for the extra casing volume so the rounds will both shoot and feel roughly the same. They are strong enough to feel like a proper .38Spl load but held back enough that I'm not worried about a chrono test for the speed limit.
All in all the bullet choice is going to be driven by what your rifle prefers. The pistols can toss any shape bullet. But the rifle is likely going to be a lot more fussy about what it eats. So in the end you need to find the bullet for the rifle and then buy lots to share with the pistols.
If you end up using your Ruger pistols for some classic bullseye shooting as a side interest then that's a whole other issue. At that point you'll want to play around with some different bullets to find something that still shoots to the same POA of your trimmed and dressed down sights but which works with the barrel and twist rate. Really though the needs for bullseye shooting and cowboy action are quite different. For bullseye shooting with a single action you'd be much better off with a Blackhawk and the adjustable sights that come with that gun so you can use any bullet and set up the sights for a mid bull aim combat style sight picture or the classic 6 o'clock "lollipop" sight picture.
