It seems like it should be easier to get to Disneyland from Los Angeles than it is, but the inconvenient fact is that unless you have an automobile, it's going to take a while - maybe as much as half a day.
Universal Studios - Disneyland
If all you want to do is see Universal Studios while you're at Disneyland, Universal sells a special one-day ticket that includes transportation from Anaheim for a reasonable price - and it's very convenient. You can buy it ahead of time at Kijubi.com.
Shuttle to Disneyland
Many of the van-type shuttle services only run between the airport and hotels or private residences and can't be engaged to go from a hotel in LA straight to Disneyland, but you may be able to use them if you try this strategy. Use one of the public transportation resources below to find out how to get from your starting point to the airport, then take a Disneyland shuttle from there.
Taking the Train from LA to Disneyland
The Metrolink commuter train runs from several Los Angeles area locations to Anaheim. Its Orange County line goes to the Anaheim station, which is shared with Amtrak. From there, you can take Anaheim Resort Transit Route 15 bus to Disneyland.
If you're taking Metrolink from other parts of Orange County, you'll have to change trains in Fullerton or the city of Orange. From their northern lines, transfer to the Orange County Line at Union Station in downtown Los Angeles. Or if you're already in Los Angeles, just get on there. Check Metrolink schedules and fares.
Public Transit from LA to DisneylandLos Angeles County Metro Express Line's #460 bus line goes to Disneyland from downtown Los Angeles. Use their trip planner to find out how to connect with it from your starting point. A trip from Hollywood at Highland through downtown and then to Disneyland will take 3 to 4 hours, but is quite inexpensive. Once you reach Disneyland, you may be able to walk to your hotel or use the Anaheim Resort Transit, which has lines that go to many local hotels.
From other parts of Orange County, try the Orange County Transportation Authority.
No matter where you start from, Google Maps may help. Input your starting and ending points, then click the icon that looks like a bus to get a route that uses public transportation.