Las Vegas is one of the youngest big cities in the American West and one of the least natural. It was founded in 1905, when land beside the Union Pacific tracks was auctioned off, and 1931 set its course: Nevada legalized casino gambling the year work began on Hoover Dam. After the war came the big hotels, and the Strip kept rebuilding itself, usually by demolishing the last version. The Venetian stands where the Sands used to be, and the Strip, about 4 miles of Las Vegas Boulevard, lies almost entirely outside the city limits.
Three days is the right length. The Strip takes a day if you are selective, because it is built to keep you inside and the free things (the fountains, the Conservatory) beat most of what you pay for. Downtown is the old town and needs its own day. The third goes to what most visitors skip, the desert. After that the casinos blur together.
Day one is the Strip, day two is downtown and the Neon Museum, and day three is Hoover Dam and Valley of Fire. The Grand Canyon is weighed honestly in the plan, because it is not a comfortable day trip from here. Start on a Monday: hotel rates tend to be lowest from Sunday to Thursday, and day three falls on a Wednesday, when the dam and the valley are quieter than on a Saturday.