A popular day trip on Long Island’s North Shore  is Oyster Bay , home to Sagamore Hill , Theodore Roosevelt’s former summer home. After touring the estate and its adjoining visitors center, have lunch in the Canterbury Ales Oyster Bar & Grill  in the village or head east another six miles to Cold Spring Harbor .
Here you’ll find a number of other good lunch spots, along with attractive shops and art galleries, a fish hatchery, the Dolan DNA Learning Center  (an arm of the Cold Spring Harbor Laboratory), and the Whaling Museum —all interesting places in which to while away the afternoon.
One of the best day trips in the Hudson Valley  is Tarrytown , home to a plethora of mansions that once belonged to famous men. Among these are the former homes of Washington Irving, Sunnyside ; John D. Rockefeller, Kykuit ; and Jay Gould Lyndhurst . Visit one of the mansion-museums in the morning and another in the afternoon, and have lunch in one of the village’s many good restaurants.
If time permits, make a stop in the Old Dutch Burying Ground , where Washington Irving found inspiration for The Legend of Sleepy Hollow, or the Union Church of Pocantico Hills , holding nine stunning stained-glass windows by artist Marc Chagall. During the summer, an especially enjoyable way to get to Tarrytown is via the NY Waterway ferryboats.
Another good choice for a day trip in the Hudson Valley is West Point , perched on a cliff at a bend in the Hudson River, and the nearby Storm King Art Center . Spend the morning at the United States Military Academy , where you’ll find a fascinating museum of military history, a visitors center, and bus tours of the campus. Have lunch at the academy’s Thayer Hotel.
In the afternoon, take in the Storm King Art Center, a stunning mountainside sculpture park featuring the work of world-famous artists spread out over 500 acres with great views. The center is only open from late spring to early fall, but offers occasional open-air concerts in its recently opened café.
Fall is an especially good time to make a day trip to the Catskills ; the foliage there is often spectacular. On your way north, make a stop in New Paltz , founded by French Huguenots in the late 1600s. The main sight here is Huguenot Street , “the oldest street in America with its original buildings”; official walking tours are offered May–October. New Paltz also holds many good lunch spots .
After lunch, continue north on Routes 213 and 209, past the scenic villages of High Falls, Stone Ridge, and Hurley . A bit further north, pick up Route 28, which will bring you into Catskill Park . Take a scenic drive on Route 28A around Ashokan Reservoir, if time allows, and then head east to the artsy village of Woodstock , home to numerous shops, galleries and good restaurants.
From the village green, drive to the top of Overlook Mountain  to explore the Tibetan monastery, Karma Triyana Dharmachakra (which has hosted His Holiness the Dalai among others), or take the steep but well-cleared carriage trail opposite the monastery to impressive views.