I live in Massachusetts and have visited the White Mountains many times. There are many interesting trails to follow. There are streams, mountains, ponds, waterfalls, and lots of trees. It's a great place to go camping or to rent a room and take day trips. I highly recommend it.
Nearly everywhere I went was free. The hike to Arethusa Falls is very nice. That's a 200 foot waterfall. The hike up Mount Washington is rewarding, but it's a LOT of work. Don't even consider it if you have kids as young as the ones in the pictures you posted.
The area around The Basin is nice (although it's not actually part of the White Mountains region).
Glen Ellis Falls (65 feet) is nice. The Silver & Flume Cascades are also very nice.
Everything I listed is free. All it takes is time and some energy to hike to. Many of them are pretty close to roads. Arethusa Falls is about 1.3 miles in, but it's time well spent. Hiking Mount Washington is an all-day hike and you need to be in decent shape. If you're not, take the Cog Railway or the Auto Road to the top - neither of those is free.
I like photography, so I'm perfectly content to spend hours at a waterfall, lake, or scenic lookout.
The state of NH has several web sites that point out areas of interest. It would be worth your time to look through them to find things that you'd find interesting and affordable.