I read that "downtown LA at rush hour" line and a literally shuddered. That would be my own personal version of hell. (And I can deal with navigating the transportation in Manhattan just fine.) Trust me, DIsney is NOTHING like this - if it were, I would NOT be a repeat visitor.
There are no transportation flyers that you'll get at WDW, but what you will find are good signs and lots and lots of friendly and helpful people. There are transportation information booths at all of the theme parks, and the guest services desk at your hotel can help you unravel any transportation problems. But I'm guessing that after a day or two on property you won't need any help at all. If you click on the little list I've made
here, you'll see how to get from park to park.
The only complicating factor is that in most cases you can't get directly from resort to resort. You're going to have to transfer somewhere. The transfer points are the theme parks and Downtown Disney, just aim for the one closest to your destination and then switch. You'll be an expert in no time.