Never flew the Hudson corridor as part of an IFR flight but I'm guessing if you request it from the controller he (she) will coordinate it for you if possible. Just don't spring it on then at the last minute!
I'm based at MMU which is about 13 NW of EWR. I typically tell MMU before I depart that I'd like a handoff to EWR for the Hudson before I depart. Once handed to EWR I'm quickly cleared into the bravo and usually told to fly directly (not 20 ft left or right!!!) over the numbers of the active runways usually at 1500ft and they'll ask my intentions. I tell them exactly what I'm looking for when heading north along the Hudson... Eg "circle the lady twice then head north to the tappanzee (aka governor Mario Cuomo bridge)" or "... depart the Hudson at the Alpine Tower and go ....." Or "...and depart or head back south and ..." Plain, clear and concise English works very well here along with using the names of the reporting points.
Keep in mind EWR has a controller position dedicated to the Hudson River corridor, I'm going to guess LGA does too. The controllers are very good so just communicate as to what you want and follow the instructions/restrictions and you'll be fine.
East River is a little trickier, you have no choice but to talk to ATC (See the Cory Lidle accident) and you can only go so far south, I've only flown the east River once and when asked what my plans were I simply told ATC "... I'd like to go as far south as you'll let me then turn back north and head back to EWR and then to MMU.". Personally I think the East River is boring :)