The idea is to get to rope drop so you are among the first people into the park that morning, thus securing very short wait times for the headliners.
At Magic Kingdom- They typically do a welcome show which is very cute and will really get you in the Magic mood! Really nice to see, at least once. A tear-jerker.
At Animal Kingdom- They do kind of a skit with Minnie checking off a list to make sure visitors have everything they need. It won't put a lump in your throat but it's cute.
At the Studios- I remember a brutally long, unfunny "skit" involving a family of the day helping to shoot a movie- that is, open the park. We were standing in a mob under the hot hot sun and all I can remember is thinking, "just shut up and let us in!" It was worth enduring it to ride TSM with no wait, though.
At Epcot- No show to speak of. Once in a while Goofy in a silver suit shows up.
Edited to add:
At MK head straight to Fantasyland if you have little ones with you. Otherwise, choose a mountain and go.
At AK- We head to the Safari, grab a FP and then ride. Afterwards, we will head to EE and either get a FP or ride, since they Safari FP window will most likely already have opened.
At HS - Head straight to TSM with the rest of the mob. Either ride right away, or stand in line for a FP. Then over to RnR or TT if you like that sort of thing.
At Epcot - Head straight to Soarin' and either ride or grab a FP. Next stop should be across the park to Test Track.
Hope this helps!