I say don't hop around the parks - it does waste valuable time IMHO. However, park hopping has been known to revive many a weary Disney walker and allow for naps!
You want to hit all of the major attractions, which is what I think is the hard part. You will want to arrive at each park for Fast Passes - and they seem to go quickly for Fast Track in Epcot. Perhaps the person who gave you advice about Epcot thought you could do Animal Kingdom in a day, and head over to Epcot for the parade and Illuminations. I agree with this strategy.
I would definitely build some 'down' time into your schedule. If your group crashes from exhaustion it won't be pretty.
You can do AK on the day you arrive, the park always seems so hot to me, so it could be a draining day. I love the Safari Ride just before the park closes but the addition of the parade has made it tough to get over there in time. I do think you will get most of your touring done if you do AK on the first day. You'll need a plan!
At the end of the day, drive over to Epcot OR MGM and see Illuminations/parade at Epcot or Fantasmic.
Day 2 - MK - arrive in parking lot 45-60 minutes prior to opening. (are there fireworks that day?) Head over to Splash Mountain the BTMR. Hop the train over to Toon Town and head to Space Mountain and Buzz.
BREAK TIME!
Finish your tour according to your family. Stake a good spot for 3pm parade, keeping in mind the floats are very high and you won't miss much by not being in the first row. If you don't want to do the parade, steer clear of the parade route!
Day 3 - Epcot (you can sleep in a tad due to later opening). Future World tour. Relax in the LAND either by doing Ice Cream Social at 3pm or grabbing light snack.
Hit the World Showcase by 4pm until closing!
Day 4 - MGM - like every other person make a dash for TOT and ride it. Grab fast pass and head to Rock n Rollercoaster. Do TOT again. Now that you are all thrilled out, you should have a good idea of what shows you'd like to see and sweat to death in. Good idea to grab fast passes for Ariel Show, bad idea to show up early for show and stand in line and be herded like a cow. MOOO.
We love the great movie ride!
Head over to the other side of the park. The Muppets are a good place to relax and never seemed crowded. We never had very long waits at Star Tours either.
I like to finish my visits with Downtown Disney the morning we leave. Hope this helps!