I don't think the lakes will be frozen enough to skate on in December

I would not suggest winter hiking beyond low elevation hikes unless you have some serious winter hiking experience. Full winter clothing and microspikes are an absolute must. Be prepared for rapid changes in weather, and retreat immediately if the weather takes a turn for the worse. Weather above treeline can be lethal that time of year, even to experienced hikers. That being said, there are tons of lower elevation snowshoe trails out of places like Jackson, Waterville Valley and Bretton Woods, and a plethora of low to mid elevation hikes everywhere in the whites. Suggest looking into /r/wmnf for more specific hiking related advice.

Skip NYC and go to Boston.

Hiking boots is probably the right move. With snow on the ground, they're the least you'll want. If there's ice, you'll probably want some microspikes. Heavier snow means gaters and/or snowshoes.

You might think that driving to NYC is a good idea, but I'd like to disabuse you of that notion. It's a long drive and the traffic will be no fun. Maybe park in Stamford or New Haven (CT) and take the train into NYC and walk around. New Haven has famous pizza, and Stamford has some great restaurants downtown (I used to live and work there).

It'll probably take about 5 hours to get from Conway to lower CT. Maybe spend a day just driving back roads to get down there; it's better than 5 hours of highway driving. If you do take the highway, I'd go over to VT and take 91 south; you'll miss Boston traffic that way.

North Conway will look nice, decorated for Christmas. If there's snow, and you like x-country skiing, Jackson is the place for that.

Funspot in Weirs Beach (Laconia) has the largest arcade in the world, full of restored classic games if you're into that, it's always fun for a few hours, plus they're open until 10pm so you don't waste your day in an arcade. https://www.classicarcademuseum.org/

On the Seacoast, Portsmouth will be decorated.

Unless you have some little New England towns picked out for your drive, you could drive to Boston and take in some sights, then take the train down to NYC.

Footwear first. I suspect that if you do any significant hiking, you will need something insulated and water proof for the 0 to 50cm of snow that could be on the ground in December.

The snow in early December is statistically not expected to be significant. But, it’s impossible to say we couldn’t have a 1/2 meter in a single storm. But wouldn’t that be awesome.

Cross country skiing is more popular that time of year than traditional hiking. Again the snow matters.

Most lakes won’t be safe for ice fishing yet. You can get a fishing license as a visitor. Conway is a beautiful area with streams and lakes in the area. It will be cold. 0C degrees during the day +/- 10.

Snow? Maybe, hard to tell. Chances can be good.

Sneakers and clothes? Anything warm and I suggest boots, especially for cold weather.

Hiking? Only if you're an expert with gear. There maybe some basic trials but if there is snow, they may have trials or even the road to the trials closed. The amount of snow that can come down can be significant.

Fishing. Maybe? Depends. Ice fishing is a thing in New Hampshire but lakes need to be have enough ice. Some communities drain the lakes in the region for the winter preparing for the inevitable freeze. Same for ice skating, though there maybe an ice skating rink around the community, you'll have check hours of operation though.

Pizza. Frankly, there is good and bad pizza to be found everywhere in New England. If you're traveling to NYC, the best pizza is believed to be in Connecticut at a place called Pepe's, IIRC. There are actually a few well known places in Connecticut and that state's sub regularly have posts about these establishments.

For Christmas events, you'd have to check around the community.

Don't waste any time trying the pizza until you get to New York City

So not go hiking in that region unless you are experienced and have proper equipment, otherwise the risk of death/serious injury is pretty high.

I wouldn't bother driving to NYC. Boston is much closer and would be just as fun for a tourist IMHO. Meredith, NH to Manhattan would take you a minimum of 5 hours of driving, plus traffic.
As others have said, weather conditions can change quickly, so I'd recommend shorter hikes with flexible clothing that you can add or subtract layers depending on the conditions you run into.

Honestly its all dependent on whether the weather wants to be nice or not, most of the time in December lakes are mostly safe to skate on, to get a official report u can just search up the lake you want to go to and most of them will have an ive thickness or a warning wether u can skate or not, and you can fish here if you have a license and they are extremely easy to get here i think like 25-50 dollars, and with hiking thats all what your willing to do most hiking mountains are open year round, and make sure u have good insulated boots cuz screw these winters lmao but have fun when ur here definitely a beautiful place to see