Hey, that's my neck of the woods! I live 15 minutes from Portsmouth.
Unfortunately, it seems as you'll be hitting our usually wonderful seacoast at it's lowest point in the season. We're between the wonderful fall foliage and the beautiful looks of snow (but maybe some will arrive just before you do!).
That said, there's plenty to do in the Portsmouth area. The Kittery outlets are right across the border, but then you're paying sales tax (ick! I love my sales tax free NH). Keep in mind, though, that where NH lacks in sales tax, we make up for in meals and rooms tax (9% on non-pre-packaged food and drink, so what you buy in a restaurant will have a tax, but what you get in a grocery store will not).
In Portsmouth itself, the downtown area, you'll find absolutely no shortage of shops and restaurants. Easily a full day or two just exploring those venues.
Your best bet for lobster might just be Newick's seafood, down the road about 8 minutes in Newington. Go around sunset--the view out the windows onto the bay is wonderful! There's Flatbread Pizza in Portsmouth, too. They have some really amazing flatbread-style pizzas (duh!) made with locally sourced, organic ingredients and they are YUMMY! MoJo's BBQ is also in town and offers some of the best BBQ I've ever had in New England. If you're looking for a bar with music, try the Press Room. If you want a bar with a lot of beer, try the Thirsty Moose (over 100 beers on tap! [it's required by state law that you include how many beers are on tap whenever you mention the Thirsty Moose

]). There's the Holy Grail in Epping (just down the road, a 20 minute drive or so) which is a great Irish restaurant/bar set in an old church. Coffee is a controversial topic, but I'm partial to Breaking New Grounds in Market Square.
Strawberry Banke is open through December and is a pretty cool historic spot. It's a bit like the Mystic Seaport museum in CT--real, authentic buildings and accoutrements dating back 375 years--and offers a really neat look at what life has been like in this same spot over the years. They do candlelight strolls on weekends in December as well so if you don't mind bundling up, that's a ton of fun.
There's the
Seacoast Science Center and the
Children's Museum of New Hampshire in nearby Dover. Both are hugely entertaining for kids. There's a few other museums in Manchester, which is about 45 minutes away.
If you don't mind a bit of a drive (about 2 hours), take a trip up to the White Mountains (my other home) and check out the outlets (sales tax free!) that are in North Conway as well as some absolutely awesome mountains. You might consider taking a trip up
Mount Washington (the tallest mountain in New England, home to one of the world's most popular skiing sites, home of the world's worst weather, the site of the fastest wind speed on Earth ever recorded by man [still bitter about the robot-recorded wind speed in AUS beating us out a few years back] and exclusive training site for the ONLY successful US Everest Expedition) via the SnowCat (the mountain is inaccessible by car from October-May and by foot only by VERY experienced climbers. The Cog does not run in winter except on rare occasions).
Slightly farther north is Santa's Village (a small, simple amusement park) which is open on Weekends. It's pretty cool if you've got younger kids. Though, I usually go about once a year myself!
In Boston, definitely hit up the New England Aquarium. Walk some (but probably not all!) of the Freedom Trail. The MFA is also pretty cool to check out. Faneuil Hall offers some good food and a historic venue. And even though it's winter, the Boston Common is still nice to walk through.
Take a drive up to Freeport and see the home of LL Bean.
There's tons to do!