I just went to Los Angeles in October from my home on the east coast for the first time in 13 years, to attend a family wedding. While there for the weekend I rode every L.A. Metro line, as well as took a round trip to Riverside on Metrolink (the commuter rail operator) over two different routes. And took my first trip on a Surfliner, between Van Nuys and L.A. Union Station.
Is it safe? Considering that the Metro Blue Line runs through South Central, Watts, and Compton, you would think not. But my son and I rode it after dark, and we felt no less safe than one would feel on transit lines here on the east coast. The same rules always apply, no matter where you go: Look like you know where you're going, and don't dress like you're rich. The only problem we had was panhandlers, but then you find them in every transit system in the country.
It was also my first time using Los Angeles Union Station (although I had visited there before just to check out the station). I don't think there is anyplace one can stand above the action. With the tracks elevated over the station, you have to have a good reason why you're going upstairs if you are not taking a train. But then, why not take a train? As long as you are carrying a Metro DayPass, you do have business on the island platform between Tracks 1 and 2 of the station because the Metro Gold Line uses those tracks. From there you can see the action on the other tracks. As for Metrolink, it is also a proof-of-payment system, so if you're carrying a valid ticket you have every right as a paying customer to be up at track level. In the unlikely event a fare inspector hassles you, jump on the nearest train.
The Metro Gold Line runs on a trestle from Tracks 1/2 towards Chinatown on its way to Pasadena. You can surely see the Amtrak and Metrolink action from this trestle. Buy your DayPass and then just ride back and forth between LAUS and Chinatown; you won't miss too much that way.
I was a tourist there, and I did have just one day to spend. I agree that Hollywood is a good place to kill time. It always was, but now that you have the Metro Red Line stopping at Hollywood/Highland and Hollywood/Vine, that area is even more accessible. I also suggest for an overview of Los Angeles you take the Metro Red Line or Blue Line to 7th/Flower/Metro Center/Julian Dixon, and then walk to the Westin Bonaventure Hotel, which has a revolving restaurant on the top level.
Two places to eat downtown: the Grand Central Market (closest to the Pershing Square Red Line station, or a mall at 7th & Figueroa called 7-Fig, near the aforementioned Metro Center station.
Between the Pershing Square station and Grand Central Market, you'll also see the lower station of the Angel's Flight funicular, which closed in 2001 when its two cars crashed into one another, killing one. I read someplace that it may reopen next year.