With four of you, the economics of the intercity transfer come into play.
1. A private shuttle will run about $100
2. The Lux Bus is PER PERSON and, at present, is $116 for the 4 of you.
Between these 2 options, the private transfer is preferred since it's less expensive and direct (you don't need to wait for others to be picked up or dropped off along the way). Add to that, the Lux Bus departs at 7:30 a.m. which might be earlier than you want to leave.
3. Rail service -- Metrolinktrains.com and Antrak.com both offer this service. One might be less expensive than the other based upon the day of travel and Metrolink offers more transfers during 'rush hours' whereas Amtrak offers more trains throughout the day (and week). For example, for a weekday transfer, Metrolink offers a fare of $9 (less, if you're pensioners). So, that's $36 for the lot of you. You could use public transit to transfer between the rail stations and the respective hotels but I would take a taxi as that will afford you with assistance with your luggage.
From Hollywood to Union (rail) Station will cost about $25; with tip, it's closer to $30. In Anaheim, the taxi ride from the Anaheim rail station would be shorter (about $12 plus tip; let's call it $15). So, that comes to $36 for the rail and $45 for the taxis or $81 in total -- a deal, imo, if you don't mind traveling by rail.
As far as where to stay whilst visiting Los Angeles, Hollywood is the best option if you'll be relying upon public transit. I live in West Hollywood and if you plan to stay on the "Sunset Strip", public transit is limited in that area -- just one bus, the #2 (and it's occasional cousin, the #302) which is a slow-going route that would require a second bus to get to most places.
The BEST way to pick a neighborhood in which to stay in L.A., is to plan your itinerary first as our city is extremely spread out and our Metrorail system (unlike London's) covers only a small portion of our metro area.
I'm a big fan of public transit but with 4 people, it gets more and more affordable to hire a car since the cost of the car is split amongst the 4 of you. Meanwhile parking at hotels in L.A. can be expensive: $30 to $40 per night is not unheard of in some areas. (Hollywood's hotels tend to have inexpensive parking or in rare cases, free parking).
Other factors which will help you decide which neighborhood in which to stay are budget and 'ambiance'. West Hollywood is a little more expensive than Hollywood and the ambiance of the two neighborhoods is different. West Hollywood is more upmarket.
Another neighborhood that I like is just south of West Hollywood. Known on Tripadvisor (in their 'hotels section') as "Central L.A." and as "Beverly Grove" by some locals, this area is bordered on the west by the Beverly Center (upmarket) Mall and the Beverly Connection (outlet/discount) Mall and on the east by the Grove (mid-range) Mall. In between is about a mile (1.6 km) of shops and restaurants of all kinds and price ranges. This is a very walkable area that isn't terribly touristy.
Transportation in the area is good: frequent northbound buses to West Hollywood and Hollywood; frequent westbound buses to Beverly Hills and (by transferring to an onward bus in Beverly Hills) beyond to the beach communities; frequent eastbound buses to downtown L.A.
Here's a cartoon map of L.A. neighborhoods that you might find helpful: