I’ll let someone else give more detail about the getting to Long Island part (when I used to make the drive 30 years ago, I always took the Verrazano Bridge across Staten Island and then the Belt Parkway but I suppose it depends where you are going and time of day).
One thing to consider is that there are two ways to get to Delaware Memorial Bridge from DC. One is to go straight up 95 thru Baltimore but depending on time of day, there can be lots of traffic, esp as you get closer to DE/Christiana. Alternatively, you can take Rt. 50/301 across the Bay Bridge and then thru the Eastern Shore to Rt. 1 which puts you on 95 right before the Del Mem Bridge. It is much more scenic and NO traffic. Depending on time you leave, you may hit some traffic on Rt 50 /w DC and Annapolis if leaving in the evening rush. In the morning, you won’t hit any. You will need EZ Pass. I almost always take this route rather than 95 thru Baltimore/DE except in the summer when there is beach traffic. I hardly see any cars b/w Queenstown and the DE line.