I’ve relied on the NYC public transportation system all my life, but now I’m in the position where I could be required to travel across Long Island.
I need to figure out what kind of car I want and what I should pay for it.
Any advice?
There is no simple answer but I’m sure you have figured that out.
If you are likely to only carry one passenger then a small car is for you.
If you are environmentally conscious then a hybrid (petrol/electric) will be a good choice.
If you are tall then a medium sedan will be more comfortable.
If in your secret heart you want to be adventurous then a SUV gives you many more roads but will be more expensive to own. Petrol/diesel, tyres etc.
There is no One Car to Rule Them All. 
autos.msn.com and carsurvey.org are good sources of info of people who have owned the cars. You can get a general idea about what could go wrong with that model year.
Don’t be afraid to test drive a lot of cars. If you are going to but a new car, test driving well-used ones of the model you want can help you see how well they’ll wear in.
When I was thinking of replacing an old car, I looked at cars I saw on the street or in parking lots and thought about which cars appealed to me.
But if you’re going to be driving from NYC across Long Island, I’d think you’d want something fuel-efficient. Beyond that, think about how much stuff you need to carry. Don’t pay for a large car if that’s not what you need most of the time.
I recently bought a certified pre-owned car from a Toyota dealer. The original buyer bought it in August 2102 and turned it in in Dec 2013 with 22,000 miles (it appears to have been used for business). He paid $32,000 new. I paid $21,500.
A certified used comes with warranty and some other perks like 1 year roadside assistance. So perhaps consider a certified used in your calculations.