This was my third time in NYC, but the very first time I've visited RI.
We were very delighted with our visit we planned beforehand, so I advise this itinerary.
Take the Astoria Line (AS) ferry boat from Pier 11 in lower Manhattan. That's the starting point, but you can also catch it from other stations in Manhattan and Brooklyn as well (2 stations on both sides - Google the stops).
However, I suggest taking it from the starting point at Pier 11 as you will be rewarded with amazing (and very cheap - 3,25 $ pp) city views from East river for around 30 min what it takes to reach RI.
Disembark at RI and explore it. Great views of Manhattan skyline, nice walk by the river to the FDR freedom park and smallpox ruined hospital. You can also take the red bus which takes you to the northern tip of RI where's the lighthouse.
When finished take the RI Tramway cable car to transfer back to Manhattan, seeing the area from bird's perspective.
Count on around 2-3 hours for a memorable visit with a budget slightly over 5$ pp.