So the long-term conflict is set up -- Lamb and Savino go chin-to-chin, scowl-wise, at the casino when Lamb tracks the murder to the head of the Savoy's credit department, where the victim worked. Savino doesn't want Lamb searching the place and accuses Lamb of trespassing in his "house," to which Lamb replies, "I am the law here, Mr. Savino, and I will decide who's breaking it."
The rest of the show is standard crime procedural, albeit one set in Vegas in 1960. The Lamb brothers ask a few key questions of a few key witnesses and listen to the words people don't say in the course of an investigation. The prime suspect at first is the victim's boyfriend, who was drunk and un-alibied and suspicious that his girlfriend was cheating on him. Lamb not only clears the guy -- the mayor has filled in Lamb's backstory as a Holmes-esque military policeman who impressed people with his solving of a series of murders in Naples, where he was stationed during the war -- but offers him a job on his ranch.
Also suspected: any of a bunch of Hell's Angels kicking up shit in Vegas, because a motorcycle tire track is found near the body. Lamb uses his Super Ranching Powers to organize a roundup, enlisting a bunch of good ol' boys in pickups, to herd the bikers up and throw 'em all in jail to stew. They are alibied because of a flower that grows a certain length of time after it rains (don't ask) but Lamb figures they saw something out there, and eventually the leader co-operates and is able to identify the murderer's car. Maybe the biker was impressed because Lamb, on a horse, ludicrously chased down the biker on his bike, zipping down the strip.
It seems that the credit manager at the Savoy was squeezing a bank worker to skim money from his employer to pay his gambling debts. The governor's daughter, who managed the high-roller accounts at the casino, found out about it, and the credit manager killed her. Lamb and his posse, manage to catch up with the guy just as he's about to fly out of Vegas for parts unknown. So the manager hops in his car and drives off. Lamb apparently tells his men, "Look, everyone but me is going to drive over here and cut him off on this lonely single-lane desert road, while I calmly walk down the road in the other direction. Then, when he turns around, you should all JUST STAY PARKED THERE, INSTEAD OF CHASING HIM, even when he turns around and starts driving back in the opposite direction straight at me. I will just calmly use my shotgun to shoot out his tires and obviously there's no chance that an out of control 1960s land yacht will hit me, so you can just hang back. It'll be more dramatic this way."