There are several ways to do this trip but flying is for me the best. Los Angeles (LAX) to LAS (Las Vegas) is a one hour flight and fares start at around $70 with either Southwest Airlines or JetBlue. If you want to enjoy the scenery, then driving or taking the shuttle is another option. The shuttle costs around $90 one way.
Driving is another option and though it takes about 4 hours, I suggest it because part of the journey will follow the path of historic Route 66 and Mojave Desert. You'll be traveling on some two-lane highways, though I-15 will take you straight to Las Vegas.
GotoBus, Greyhound and Megabus all offer bus services between Los Angeles and Las Vegas. Megabus does offer the odd $1 discount, but pretty all three offer similar pricing. By the way, Amtrak also runs a bus from Los Angeles to Las Vegas.
I traveled by car to Las Vegas and I certainly recommend this form of transport, because it includes passing the famous Route 66 road. The advantage of this type of traveling is that you can see a lot of destinations and see a lot more than when you are traveling by plane.
Drive if you can, otherwise you can travel via bus too, it is not that uncomfortable when traveling by bus as most people think it is.