be careful not to overwhelm yourself. ive done a lot of traveling in my life and i can confidently say that the places i liked more were the places i felt comfortable because i planned it out and felt in control. the places i identify negatively with are places i didnt have control. and none of this has to do with the quality of the place. ex: i hated a wonderful place like naples because i had no idea where the fuck i was going. i loved staying in a sketchy place like the red light district because i felt prepared and knew the area and the customs very well prior to going. all i can suggest is making sure you have adequate time between each trip and school, or you're better off not forcing yourself to attend the ASW and just going at your own leisure.
for nyc: fly into jfk. its the most convenient with the subway straight into the city. if you definitely want a cab, LGA is good too. get there a night early perhaps if you can afford it so you aren't rushed or pressured. you'll feel safe I promise. I did the trip by myself for the first time recently just to visit NYU & Columbia and it was very simple/affordable if you use jetblue. take the airtrain to the jamaica subway station and then go into the city via subway. cab is easy but 20 times more expensive. pro of a cab is you get to see more of the city. con of the cab is you dont really get the feel for the subway, getting around, which is HUGE when you consider how you will tolerate 3 years of this city. stay at a hotel near NYU or midtown, but not too far towards the west or east side. youll feel more comfortable the closer you are to the center of the island. use your free time to check out things like times square, rockefeller, central park (not that great this time of year compared to others), walk around chinatown or the village during the day if you have time, use the asw for nyu to go out in the village at night. columbia and nyu are not more than 20 min via subway no matter where you stay so youll be ok.
for boston: don't know it very well. from what I know, cab is generally easier. i believe you still have to take a bus to the boston subway which is probably worth more than the hassle. Cambridge is NOT intimidating.
for durham, i just visited it last week on my own accord. cab will be expensive and its not a city you want to necessarily be stuck in w/o transport. car might be preferable unless you are a nervous driver or something. absolutely wonderful campus once you get to it from the airport. i suggest driving to the city instead of flying there if you live close enough, but i dont know how parking is. southwest into there is very cheap as well.
edit: saw you were concerned about money. nyu offers a travel stipend if you email their office and have you financial aid stuff submitted already. also, if you join the nyu class or 2014 fb group they can arrange for you to work something out to stay with someone, just make sure you convey its importance... but dont ask unless you plan to do both days of the nyu asw...
honestly, if harvards on the table i would limit it to that and the ny schools, unless duke is offering VERY significant money. i would have a hard time even going to the asw there if i was in at harvard... and i personally just wouldnt go if i had so many choices and schools to visit unless they were offering a LOT of money.