OK -- some assumptions:
1) Prevailing winds from the SW, 5-15k, with little wind in the morning.
2) Work the currents and winds, just about anyplace in LIS has more than 1-1.5 + knots peak current.
3) Summer cold fronts will be preceded by squalls (some dangerous) then NW winds
4) Winds with an easterly component accompany foul weather

Here's how I'd do it:
When you consider #1 above, you're more likely to have favorable sailing winds if you go from west to east. The other way (heading west) is more likely to involve extended motoring.
Coming from BI -- it's a perfect reach in SW winds from the entrance of Salt Pond at 300-degrees magnetic to Watch Hill Passage. Follow the buoys to Watch Hill and anchor behind Nappatree Point or head into Stonington CT. I recommend the latter. Anchor at the south end of the harbor or grab a mooring with Dodson's Boatyard. Stonington is a charming colonial-era "borough" with beautifully preserved town homes that has a number of truly excellent restaurants, a few nice boutique shops and a terrific nautical shop with antique art and ship collectibles (entrance off an alley -- ask a local). It's a short cab ride (about $10-12 + tip as I recall) to a supermarket and downtown Mystic, which is another great place to walk around. Stonington/Mystic is a great place to spend a day or a week.

From there, if you have NW winds you can sail to the Long Island Forks (Gardiners Bay) which I wrote about below and do recommend, or if you want to go West I'd recommend the CT River as the next stop. If you go up to Essex I recommend you spend a few hours at the CT River Museum. A small selection of nice shops and restaurants in Essex too. If the LIS forecast is typical light SW or no winds, I recommend when you leave Stonington through Fishers Island Sound you motor SW and go to Mattituck NY. Mattituck is a small well-protected anchorage at the end of a very pretty (short) river. Launch the dinghy and go ashore at the small dinghy dock -- it's a Federal park and they have showers (open until 7pm if memory serves) then walk down Love Lane about 200 yards into town where there is shopping, restaurants and gourmet shops, etc. A very pleasant stop. Be prepared for thick stinky mud when you raise the anchor. A reason I suggest Mattituck is because (with continuing SW winds) it positions you for a nice sail on a reach back WNW toward the CT shore.

If I continue this I'll be writing my own cruising guide, so I'll cut to the chase and give you my recommended and not recommended lists:
Recommended:
CT River, all - I can give specific mooring recommendations if you ask
Westbrook (Anchor in Duck Island Roads).
Clinton (Cedar Island marina has a good restaurant right there).
Branford harbor -- I anchor north of Big Mermaid but that area's filling up with moorings, or Brewers is a good marina but high prices (slips are $$$).
Milford CT (Milford Landing marina is transient only, and right in town).
Port Jefferson NY (A+ shopping and restaurants galore). Get a mooring at Setauket Yacht Club it's right at edge of town and they have launch service. You can also anchor in the north end of the harbor.
Northport NY (Rated A++. Nice restaurants, shopping. Take a mooring with Centerport YC or Seymours and take their launch to town.) Lloyd Harbor is also nice, and Northport Bay has a good anchorage at the west end protected by West Beach (south of Eatons Neck)
Oyster Bay (Sagamore YC is preferred over Oyster Bay Marine center for moorings and launch, but you can anchor East, West or North in Oyster Bay and it's all good.)
Manhasset

Not recommended (this is not a list of places to avoid, its a list of places that are not as good as others for a cruiser in a sailboat):
New London
Niantic
Thimble Islands (although the cruising guides talk it up, it's an exposed anchorage strewn with cables and boulders).
New Haven, Stratford, Bridgeport (although Black Rock Harbor west of Bridgeport has a nice yacht club, and Captains Cove Marina is quite a scene if you like it loud).
Norwalk is not sailboat cruiser-friendly (though it has a nice aquarium with IMAX and a lot to offer for dining in that neighborhood, but you'd have to take a cab or walk through some sketchy areas. The marinas in walking distance are only for stinkpotters and the public docks are not cruiser friendly. Norwalk Yacht Club is nice, but isolated. The Norwalk Islands have nice day anchorages.
Huntington Harbor (If you have limited time, Northport next to it is way better.)
Stamford
Greenwich, although the Greenwich Islands are a nice anchorage in settled weather they are exposed. Many people use them as a day anchorage and it empties out at night.
Port Chester
Hempstead (skip to Manhasset which has more to offer transients ashore)
If you didn't go to CT your next stop on the LI shore from Mattituck is Stonybrook, but I recommend you skip it and go to Port Jefferson.
City Island -- it's charm is a memory reserved for those who know it.
Little Neck Bay and Flushing Bay
Rye (it' just YC guest moorings no room to anchor and nothing within a close walk)

There are a few I didn't mention because I'm on the fence about them, as the towns are close and have something to offer but the facilities aren't cruiser friendly (e.g. New Rochelle)