My suggestion, especially if the first time family camping in NH, would be Hancock Campground NF east of Lincoln on Rt 112 (The Kancamagus Highway). The Town of Lincoln is nearby (touristy) and the Kanc has much to offer for hikes of all levels. The CG sits along the Pemi, so water to play in! Up the road is a wonder hike (The Wilderness Trail) along the Pemi and up to several waterfalls (13 Falls Loop). A day's loop drive can bring you around Mt Washington. Temptation would be to drive up Mt Washington (which as a resident I still have not done), yet I will suggest that going to Fabian Station and riding the Cog is a better choice. You, the driver, get to relax and enjoy as well! The Flume (and other attractions) are just up the road (Rt 93) from Lincoln as well.
Traveling west on RT 112 (as travelinjohnnie suggests) leads to a lesser traveled area on NH. I would have suggested this as well, but... one of the best drives over there is closed down due to a washout (s) in the road (not Rt 112, but another very scenic road).
Do hope you enjoy my state! Feel free to PM me for any info I can pass along!