There are a few ferry ports on LI. I am focusing on the Ferries that transport people to and from Fire Island. The three I have used is the Davis Park Ferry out of Patchougue, the Sunken Forest/Sailors Haven out of Sayville and the Ocean Park out of Bayshore (pictured). They are all about 12.00$ roundtrip and if arriving by car you have to pay for parking.
http://www.pagelinx.com/fififerry/index.shtml is the link for Bayshore Ferries
http://www.sayvilleferry.com/ is for the Sayville ferries
http://www.pagelinx.com/dpferry/ is the link for Patchogue
The Long Island Expressway runs the length of long island up until the split of the tail. It's exits progressively get higher as you head east ending at exit 70 something which will drop you into Riverhead. Each exit will drop you into a town unto itself. Exit 68s is where I grew up and ends up being my own most used exit. Traffic can get horrendous on the week ends, rush hour and holidays. The full extent of the LIE should take about 1 hour and 45 minutes but due to heavy volume has taken me 3-4 hours. It starts straight out the The Queens Midtown Tunnel and just heads straight east. The speed limit is 55mph but not one person abides by these rules if there is open road. Try travelling at night or obscure hours because this strip of pavement is jammed.
Trains from New York 34th Street, Penn Station and Brooklyn, Flatbush to all over Long Island. You can get price and schedule information from MTA website:
BTW, you better buy your ticket before boarding the train since they charge extra for buying ticket on train.
A taxi might be expensive, and you would have to call constantly, which will become very costly. Renting a car would be the easiest because there really is a lot to see, and you would constantly have to get into the car to see it anyway because there really isn't anything within walking distance you would want to see, except maybe a store. The train is also expensive from town to town, so a car is your best bet.
You can take the train to most beaches in LI. Go to the website there are some special packages to get out there.
To get Islip Airport in Long Island, you can take LIRR to Ronkonkoma, there is bus service to the airport.