Sunday mornings are a great time to make the trip to the East End, but remember that there is a big difference in amount of traffic depending on your time of departure. If you can leave as close to 8am as possible, or even before, you should be in for a stress-free trip. By the time 10am rolls around, beach traffic can be pretty heavy.

You should double check but if I recall correctly, you would take the Throg's Neck Bridge to the Cross Island to 495/Long Island Expressway, and head east. (I often make the drive to the East End early Sunday mornings, but I depart from Manhattan, thus the need to double check the access to the LIE from NR)

On a Sunday morning there should be little traffic out to the Island - certainly IMO not enough to warrant taking the ferry from Bridgeport unless you'd like the ride.

There's basically only one route you'd take to get on the Island - you can google it and see how straightforward it is. Google gives 3 routes once you're on-island, with estimated current traffic times for each. (I-495 is the only one of these routes that allows trucks but it's also a much faster, multi-lane roadway; the Northern State and Southern State are scenic, winding parkways from Robert Moses' time, though as major routes, they still are pretty fast.)

Oh, and I'm assuming you mean Southampton New York. You didn't specify, and you could just as easily have meant Southampton, Massachusetts, another lovely summer destination.