I took Greyhound out of NYC last year for interviews in Philly, DC, Boston. It was fine, no problems. I've taken Greyhound many, many times out of NYC without complaints. Tons of people ride the bus and tons of buses are always leaving to/from these cities b/c they are very popular routes. You could take the Chinatown buses, but they really aren't that much less expensive than Greyhound. I choose Greyhound over Chinatown since they are national and have tons of choices to get you where you are going. Plus they are easy to find in Manhattan - go to Port Authority, a huge place, can't miss it - vs. Chinatown where you line up on a street corner. In Boston, however, the Chinatown bus leave right out of South Station right next to a Greyhound bus, so I guess it would be ok coming out of Boston.
Amtrak is a nicer ride, but it's always much more expensive than Greyhound. For example, I am going to Philly next weekend. Greyhound is going to cost me $32 ($14 each way + $4 ticket fee) and I can choose whichever time I want to get on the bus. Greyhound takes 2 hours door to door. Amtrak's cheapest option is $106, and it doesn't even have the times I want at that price. Amtrak takes 1.5 hours. Ummm, for an extra 30 min and 1/3 the cost, I choose Greyhound.
As far as renting a car, who cares? It is possible to arrive on time at a place using public transportation. If in a bind, I'll go to the front desk of my hotel and call a cab. Renting a car just to show you are "reliable" is ludicrous. Just b/c I don't have a car now (a car in NYC is a big expense and really unnecessary) doesn't mean I can't buy one when I match into their fabulous program in the middle of America.
Although I'm a fan of Greyhound out of NYC, I have rode the Greyhound out of Houston to San Antonio and that is a trip I will NOT do again. 😉
FYI, if you're going to do Greyhound, buy your tickets online b/c they are cheaper than the "walkup fare" you'll pay to buy it in the station. And make sure you look at the "special fare" link (whatever it's called) b/c that where it links you to the cheapest prices for the most popular routes.