I'd recommend trying out these two different routes on days where the traffic would not be heavy near the coast (such as this weekend with these fantastic beach days). Best time to scope out the beaches are weekdays or anytime during the fall or spring.
1. Route 16 south to Portsmouth and take 1A over to Rye, North Hampton and Hampton beaches. Great Beaches!
2. Route 16 south to Sanbornville. Take Route 109 through Acton and Sanford over to Wells, Moody, Footbridge or Ogunquit Beaches. Wells and Ogunquit have the most public parking. All of these beaches are great. York Beach as well.
Either way it will be 90 minutes or so from Tuftonboro.