astrein, I don't think it sounds like you need a high end or necessarily new bike, but a lot of what you need to look into is the mechanical condition of the bike. If you end up with a bike in poor condition it could be unreliable and you may spend more money than you expect on repairs if you don't learn to do them yourself. Fortunately, you live in an actual city so there's a bicycle cooperative in your area
http://times-up.org/index.php?page=bike-co-op . They can help you learn to work on your own bike, and have tools to use, which will keep your total cost of ownership down. Might also be worth stopping in to see if they have any used bikes for sale. Coop bikes often are in need of repair, with the expectation that you'll do that repair at the shop (with their help).
Both those bikes are honestly kind of crappy, although the Ross is appropriately priced for it (the Schwinn is the better bike). Ideally look out for name brand bikes from the 80s-90s. Japanese brands road bikes from the 80s are often good values. Rigid (no suspension) mountain bikes are also legitimate options. Avoid anything sold at department stores or Walmart etc, like new Huffy and Pacific, and Mongoose bikes, these are crap.
Also, I haven't been to NY sadly, but my understanding is it's fairly flat? If so, a singlespeed (or fixed gear, you can switch to a freewheel on most of these anyways) might be a good option for its low maintenence. These are trendy so retain value well, but are less expensive new than bikes with derailleurs.
BigJeff's recommendations are in my opinion ridiculous considering the nature of what you're asking. Those are all nice things, and I have them, but clipless pedals and shoes are entirely unnecessarily for transportation (though nice!), $20 hemets from Walmart/whatever are fine (slightly heavier, slightly less ventilating, exactly as safe, maybe not as good fitting? I have both and the cheap Bell just looks a bit bulkier, is actually about the same weight and fit). You should buy a floor pump though, though they're available for less than that. A mini pump, patch kit, extra tube, and the knowledge to use them are actually pretty key. And you definitely should NOT cheap out on the lock, get a strong U lock (no cable locks! maybe heavy chain locks) and be willing to pay for it, thieves will absolutely steal low value bicycles if they're easy targets. Also get lights, they help cars see you and consequently not kill you at night. Cheap front lights are sort of weak, so if you actually want to see anything in the dark consider spending some money (there are good chinese lights cheap on amazon and dealextreme, also).
Good luck finding a nice bike.