Why not one of the established ranges? A bit of a freeway drive, but take the 95 on past Henderson and you can choose between the Desert Hills Pro Gun Club and the Boulder Rifle & Pistol Club. Both are open to non-members, but the latter only on Tuesday, Thursday, and weekends.
Not free, but you have some amenities. Bathrooms, shade, tables, target stands, etc.
You'll probably want to join after you get used to it!