I did the reverse route, going from NYC to Philly via Trenton. Between Newark and Princeton we found the
East Coast Greenway (ECG) to be, for the most part, well-marked, and it took us on nice, quiet roads. There were a few exceptions with missing signs and interim routes on busy roads, but it was quite nice. Take some maps and cue sheets and you'll be set. (Not to mention better prepared...we had no cue sheets and just one iPhone. Whoops!
)
Trenton to Princeton along the canal is great, and you should be fine on your Schwinn as long as you're not gunning it and giving yourself plenty of time.
PATH from Newark is a very good option, but it *is* possible to go from Newark up to the GWB if you're really determined. It'll get a bit hilly on that section.