Thanks everyone. Just to update this for anyone else wanting to undertake this folly: it is the G-LAX shuttle bus that runs out to the Aviation station. You get it from the Blue Airport Shuttles bays on the lower arrivals levels. [You could take a B-Parking Lot shuttle and walk 5-minutes but not really worth it unless you want to look at the collection of exotic aircraft on the lawn of a local banquet hall: WWII fighters of all origins, some Korean War jets and even a Bell X series rocket plane!]

However, once at the station, the ticket dispensers only take cash [coins or $1 bills]. Best deal, unless you are just going to a city hotel, is a $3 day pass [$1.50 if you are over 62]. The Green route is all above ground and runs along the right-of-way of a major expressway. Service is about every 15-minutes and you will likely want to transfer to the Blue Line at the Rosa Parks station to get into the city centre. [Very poor signs but walk to the east end of the platform and take escalator or lift down to the Blue Line station.] This line also runs on the surface, but is more an LRT going through Black [great view of the Watts Towers] and Hispanic neighbourhoods. At 7th station you will likely want to change to the Red Line, a true subway which runs either to the Union Station or out in two spurs to Hollywood.

Total travel time between LAX [hotel shuttle to Green Line shuttle to station to Wilshire/Western station and transfer to bus for Beverly Hills, Museum Mile, etc.] is about 2-hours. However, as a tourist is does give one a peek at an LA that most seldom see. Once you figure it out, the LA Transit System is actually quite good and very cheap. There are also buses that go right into Santa Monica from the LAX transit centre, which also makes for an interesting drive. Just $1.75 each direction.

Biggest drawback is that the whole system is automated -- except for the drivers -- and I encountered just one fare checker in two days of riding the system, other than buses, of course. Thus it is impossible to get directions, etc. There is, however, one can get information and a very good transit/city map at the ticket kiosk at Union Station. However, you must take the escalator up to the "Transit Centre" there, and not the railway station at the opposite end.

This system is also known for its interesting public art at all stations. This also makes for an interesting "tour".