The only good summer fun this state offers is the White Mountain (ie large hill) climbing/hiking. Or the lakes, of course, where they never should have allowed power boats but, freedom. Those mountains are all good hikes with no climbing gear needed except an intrepid spirit, sneakers, and pretty good health - and do not forget to bring your legs. You might need them. Nothing wrong with trail maps either, and some snacks.
NH has 48 4000-footers, and the outdoor people from Boston, New York, CT, etc. with lake or ski places up there tackle each one. That's the main summer outdoor activity up there. Insanely vigorous people try hiking those peaks in winter. I tried that once, in college. Trust me. It is not fun but it is a good test of manhood.
OK, there are some fun trout streams too. Rumor has it that there are some wild Brookies, but I doubt it. Not much else to do there for fun and recreation in warm weather. The winter skiiing is good, though, and in the fall the woodcock and grouse hunting up around Pittsburg, NH, is quite fine. A guide is a good idea for that.
Other than that, Live Free Or Die. Thus lots of meth labs and lowlifes in the woods. Not much cultcha if you go for high cultcha.