You're from Europe, absolutely no need rent a car, go all the way up to Littleton to visit Schillings. Some suggestions as you go on your way:

Some genuine New Hampshire/American spots:

Stop in Manchester, if you're interested in American politics or food. If you want a classic food experience, go to Red Arrow Diner. Absolute heavyweights in the political arena, and a damn fine diner meal. That being said, if you want to find the quintessentially American food experience, go to the Puritan in Manchester. Get the tenders. With the dipping sauce? Best chicken tenders I've ever had. Want to be a real American? Get the landslide.

If you want an truly quaint New England/American diner experience, stop at Joanne's Kitchen in Nashua. Some of the best diner food I've ever had, some of the most satifisying meals I've ever had.

In the northern Lakes region? Cross a couple minutes over the state border into Maine, to go visit Kristas over in Cornish. Hands down, some of the best meals I've ever had out. Superb experience.

One more: if you're really going to Gilmanton/Gunstock -- the Tilt'n Diner. Personally, I haven't been back there in a couple years, but the food absolutely slaps every time I go.

DM me if you have any more Qs!