I know some folks shoot skeet at
Nellis Airbase but I don't know if they have any sort of pistol range there, it might just be clays.
I'm on the southwest side of town and drive to
Desert Sportsman's Rifle and Pistol club. I like that range because they have many, many competitions like IDPA, benchrest rimfire, silhouette, and more I can't remember. They have about 20 pits that are 50 yards deep with high berms for close range work. There are also 100, 450 and 600 yard ranges and a shotgun area.
On east side of town, there's
Boulder Rifle & Pistol Club. They actually have a courtesy range where you can shoot even if you're not a member. I'm not sure of the prices for that. Boulder is a also a nice facility, high berms, plenty of room, it's just too far from me. Suarez International and Fight Focused Concepts teach some of their classes out there.
Don't forget about the
Clark County Shooting Park at the north end of the city. It is way to far for me to frequent with any regularity. I only tried their skeet range and it was very very nice. $6 a round, IIRC.
When I want to get away, I drive into Nye county (west towards Pahrump on 160) and take Wheeler Pass or Trout Canyon. There are also some shooting holes in the National Forest before Pahrump, at Lovell Canyon.
There are some indoor ranges in town but I refuse to enter them except to qualify for CCW. Too many Mall Ninjas and tourists. I believe American Shooters on Arville has an indoor version of IDPA which appeals to some people since there's a time commitment to get out to the desert or to get to one of the ranges, all of which are on the edges of the town.
Your ear protection is fine for outdoor ranges. Some people like to double up and put the muffs on top of the inner-ear plugs. I just use the foam ear inserts from Home Depot.