I used to go twice a year to specific private land properties near Clayton, NM and in the TX panhandle on the border between TX and NM. I never used any guides; we just asked around and made friends with landowners who wanted them eradicated and shot for free. We found the best time to go was in the spring…April through early June. We saw more activity in the spring than any other time, when the grass was greening up and lush. Any earlier and it was too cold; much later and the heat was too unpleasant.
We would bring shooting benches, canopies to give us shade, lots of coolers full of refreshments, optics, 2-4 rifles apiece and LOTS of ammo. I used to bring 3000-4000 rounds of ammo just for me for a weekend of shooting, and half the time, I would use all of it! As stated, it isn't "hunting," it's sniping big rats. On our better places on good years, we could setup in one spot and have plenty of shooting all day long without relocating!
As for rifles, I would bring varmint-style heavy barreled bolt actions in .223, .22-250, .204, 6BR, 6PPC, 6-.284, etc, usually with 4-16x or 6-24x class scopes. You can really burn up barrels on rifles on a good place, which is why I'd bring 3-4 rifles and rotate them. The biggest expense was ammo, and you'd better handload, or it certainly got expensive!
I stopped going because the landowners I was in contact with either passed away or sold their property and the fact I was having greater difficulty getting enough brass, powder, and primers, and it was getting very tiresome and very expensive to load so much ammo.
But, it was fun!