there is a direct bus from Anaheim to downtown Los Angeles
the cost would be $2.25 for the trip or you could get a day pass
for $5.00 +1.00 for freeway zone charges)
the day pass would be good all day including the subway if you wanted to go Hollywood or Universal studios
another alternative is to use Metrolink from the Anaheim Metrolink/Amtrak station to Union station (hold on to your metrolink ticket as it acts a like pass on connecting transit
(including the subway to Hollywood or Gold line to Pasadena
There are several ways to get to the _touristy_ areas of Los Angeles from Anaheim. It_s only about 30 Km. or so. There is the city bus system, a commuter train, and commercial guided tour companies that might meet your needs. I know you said you didn_t want to rent/hire a car, but that would be the most convenient means of travel. Traffic laws are fairly common world wide, and the traffic really isn_t that unbearable (or maybe I_m just used to it). Welcome to the United States and enjoy your stay.
When it gets closer to your _special_ day post this question again, and we can give you more specific ideas.
2. take private shuttle ($70 approx).
3. not sure about trains, but try amtrak from anaheim to union station.