I've done this drive many, many times.
You are fine to go 80 from LA to 15, if you don't people will probably hate you. You may hit snow going through cajon pass or crazy traffic if you are doing the drive on a weekend. You can honestly sit for hours on the 15 before Cajon pass if you don't leave at the right time of the day.
Cajon pass to Vegas, I'd stay just under 80. A lot of cops chill out in the CA section. Beware.
Vegas to Utah is quite beautiful. Slow down as soon as you hit Arizona though, usually lots of cops pulling people over for the 20 miles you drive through there. Once, I got a warning for going 4mph over.
A lot of Utah is 75/80mph or test zones and what not. You can probably go a bit faster, but I wouldn't press my luck too much since the speed limits are already so high.
The drive usually takes me around 13 hours with stopping and so forth and I almost always have done it by myself. Just make sure you stay gassed up in Utah, there is a 111mi stretch or so with absolutely no gas stations. During this stretch, people also seem to drive as fast as they want, be careful though and watch the signs cause you can overheat, run out of gas, or take a turn too fast and have that be the end of your journey.
