Welcome to New Hampshire! As other posters have mentioned, no sales or income tax, but the property tax is steep. Remember that when you buy a place.
Portsmouth is a nice little city. I don’t go there frequently, but I have enjoyed the Coat of Arms Pub, and the Portsmouth Brewery; both have good beer and good food. There’s another place I have been that was very good, but the name escapes me at the moment. The seafood in Portsmouth (and most of NH) is great; try the chowders or lobster. There are quite a few shops and restaurants in Portsmouth, and of course lots to do around the area. The ocean is right there, and the White Mountains - which offer some wonderful hiking and sightseeing - are only about an hour away. I would suggest living closer to Milton if possible; I wouldn’t want a 35 mile commute in the winter here, but of course YMMV (literally).
Hampton is the “tourist” town on the coast. It’s loud, filled with interesting people and shops, and always busy during the summer. The Hampton beach casino is one of the best venues I have ever been to for concerts; it fits about 1,000, so it’s very intimate. Enjoy Hampton beach when you can, but I don’t “beach it” there. There are far better, less busy, and more importantly, cleaner beaches to enjoy. (If you like fishing: fresh water fishing requires a permit; salt water fishing does not.)
For night life, Manchester offers some adventures. It’s the biggest city in NH but is still quite small - something like 130,000 if I recall correctly. Quite a few bars and restaurants, many of them international. There’s a great Brazilian Steak House there; I’ll try to find the name. The local sports teams have two great arenas: the Verizon Wireless and Merchant’s Auto Stadium. I would highly recommend either a NH Fisher Cats (minor league baseball) or Manchester Monarchs (minor hockey). The tickets are cheap ($10-20), the atmosphere welcoming, and the play exciting.
I lived in Hanover for two years as well, and there are quite a few good restaurants and things to do up there. Dartmouth brings in a considerable amount of performances, speakers, gallery showings, and the like. If you are up there, have breakfast at Lou’s. It’s fantastic.
Boston is only an hour or so away from most of southern NH, and is another great city, but I’m sure you know at least some about it.
The best advice I can give you: when you first get here, take the state highways around on the weekends. I still do this occasionally, and am always finding new places and people, and I’ve lived here most my life.
Feel free to PM me if you have more questions!