I guess it depends on what you want out of the trip. If you've always wanted to travel by rail, I think you should do it. On the other hand, you get a lot more value for your money by flying. Following is one of our typical experiences. We found that it's better to pay incrementally more to add a second air destination that augments our travel. We've done it for most of our family vacations and I find it to a great way to get the most out of a vacation.
About 7 or 8 years ago I made reservations to travel to Glacier National Park via Amtrak. I thought it would be a cool experience, especially since the kids were about 8 and 10 years old. The week we were to travel, Amtrak was in a financial crisis and may have closed for a while, so I rejected that risk and canceled our reservations.
I then called a travel agent and asked how we could do it by air. She said we could fly into Kalispell but there would be a layover in Seattle. I asked if we could have a two day layover in Seattle because I thought it would be cool to add that to the trip. I had to pay extra for that, but we did spend two days in Seattle before going on to Montana. It turned out to be a great trip. For not much more than the price of the purely Montana trip, we got to see Seattle's China town, aquarium, the Sound, pacific rain forest, tidepools, the Olympic peninsula, Seattle's ferrys, etc. It really added a dimension to the trip. I think it added about $500 to a two week trip for four people, which isn't bad at all considering what we got.
I'm glad we changed our plans because we got a lot more from the trip than we otherwise would have.
At the time, train schedules were highly variable with potentially long waits in lots of places. The costs were higher than air travel (and I believe it's still true today.) By all accounts, the system is pretty rundown today, and you should discard any romantic notions of train travel that you might have.
If I were going on your trip, and I don't know where you live, what I would consider doing is:
1) Fly to Las Vegas and enjoy the sights. It's not all gambling. It's an amazing sight. Just going to Caesars Palace and walking around is an amazing experience.
2) Arrange to rent a car in Vegas and then return it in Salt Lake. This is do-able, but is tricky with online reservation systems so you need to do it over the phone. Travel agents are good for this kind of thing, even if you have to pay a $50 service charge for making all the arrangements.
3) Drive from Vegas to Hoover Dam south, to see that great sight, but stop and wade in the lake by the Marina first and enjoy the cool clear water at the beach there.
4) Drive north to the NORTH NORTH NORTH rim of the grand canyon. This is much nicer than the south rim, and it's on the way to Utah. You'll drive over scenic high desert and through the beautiful Kaibab forest.
5) Drive on to Salt Lake
6) Fly home from Salt Lake.
Spring (i.e., before mid-May when it gets hot) will be a great time to do this.