Well, 4 hours from NYC... That could get you all the way to Providence.
two aquariums... http://www.maritimeaquarium.org/home.asp http://www.mysticaquarium.org/
I haven't been to Maritime in Norwalk but the GF says it's better than Mystic... but I find that hard to believe. Also there are some mind blowing restaurants around the Maritime (not sure if you're a foodie).
Then there's New Haven, museums (art, history, natural history, ect.), again great restaurants, Sarah T. Crosby Conservatory, Community Greenhouses and Gardens and my ugly mug.
Providence... well that's just a fun town all around. You got Roger Williams Botanical Gardens not to mention the zoo. Again, great restaurants. In the summer there's usually always some street festival that ends up being pretty cool. There's also the Firewalk which is pretty cool.
Shoot, now I want to go to Providece today.
I'm a 45 min trainride from Penn Station, NYC and I have a pretty nice collection of some really nice plants... but I don't think I'll be around much this weekend (GF's family reunion Sat & beach trip Sun)
Schloaty's in the area too, but upstate instead of on LI.
As for things to do and places to see? Well NYC has it all! Except CP's...
The NYBG is really nice! Not much in the way of CP's though... Brooklyn BG is nice too and I think they may have a outdoor bog garden these days.
The AMNH is always incredible and the MET is straight across Central Park with the Guggenheim up the street from there or the MOMA down the street.
There's Central Park as mentioned or you can take a train up to Riverside park and walk all the way down to the Battery along the new Hudson River Park with a very slight detour of the Hi-Line Park along the way.
Coney Island Aquarium and Cony Island itself. There's the Riverhead Aquarium on Long Island which is pretty cool and has some nice outdoor CP Bog Gardens... They are very involved in rescueing local marine life.
Just a few thoughts....
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Bonus tip: in the Reptiles & Amphibeans wing at the AMNH, there are S. Purpurea as part of the diaroma in one of the exhibits
The Pine Barrens are close maybe 2-3 hours away. Byre State Park and Wharton State Park have trails where you can see carnivorous plants in their natural habitat. Wharton is larger and has more trails, check out the Atsion Lake campsite area. elgecko (Steve) has some great pics on some of his pine barren trips (9 so far) he has posted here. There's also the infamous Jersey shore. Point Pleasant and Seaside Heights about 2 -2.5 hrs from NYC have boardwalks with all the food and game stands. Sandy Hook is much closer (45 min from NYC) is a Nat'l Park(Gateway Nat'l Park) that has some seaside wildlife areas and swimming areas.
I'd second quogue's recommendation of checking out the NYBG or Brooklyn Botanical Gardens. Not much in terms of CP's, but still sites to see for sure. Gorgeous grounds in the heart of millions of people! And being so close to NYC, it'd be quite the easy trip!
I've heard great things about the NJ Pine Barrens from RL3786, but its vast land for sure....
I've also heard wonderful things about the Long Island Pine Barrens which would probably be about a 2hr trip or so and I'd imagine it isnt as vast as NJ considering its size, so easier to explore.... lots of drosera and sarrs from what I've researched.... shame I haven't explored myself to be honest....