Central Park, nice. I've done many laps there - it's a beautiful loop. Nothing like the smell of horse manure at 6:30am while gulping down large mouthfulls of O2. It's even more fun racing in the rain. You learn to keep your mouth shut real tight.
If the dealer is legitimate buying online is fine. If you go this path, when the bike arrives you will need to install the wheels, the seat/seatpost, and the stem (the handlebars, tape, cables, and shifters will already be installed). Depending on who did the initial install on the other side the bike may be ready to go with gears perfectly adjusted. Installing the wheels is a matter of understanding the quick release mechanism. Attaching the stem is a little more tricky but with all the vids on youtube available and a very basic mechanical inclination it is actually pretty easy. You should be able to get the whole thing done with just a couple allen keys.
If the initial install was sloppy you may need to get a very basic tuneup at a shop (should be just derailleur adjust and a once over) which should cost less than $50 (hey it's the big city) and ideally should be closer to about $20.
These days most bike shops are good and knowledgable. There will be deals on last years bikes - usually to the tune of 10-20%. In the city I like Bicycle Habitat. Sids bikes is a good outfit too. it doesn't sound like you are attached to any particular brand which would limit your search to paricular dealers. In Brooklyn, 9th Street bicycles is a good shop - I race for them.
If you order online you should have an understanding of what size you need. As sizing conventions vary, you can most easily use "effective top tube" length as a measurement. Try a couple bikes local, check the sizes, find their corresponding top tube length which are always available on a company's website (or at the shop), and match it up to the top tube length of the bike you are looking at. If not familiar yet with your own size, it is always better to test ride a bike with similar measurements before pulling the trigger.
At 6ft I may go with the 56, but another reason to be real sure about your size is that that is an old school horizontal top tube (as opposed to "sloping top tube", and which usually facilitate a more comfortable riding position) and therefore has a short "head tube" (compared to the "sloping" sizes). The 59 may be too big and have you too strectched out. The traditional 56 may have you too low.
