I own there and have been going to NH for years. Our son lives there as well. Definitely drive the Kancamaugus Highway and stop along the way. Plan on a whole day for that. You can also visit North Conway when you get to the end.
Would need another day to do some of the things in the North Conway area, like Dianas Baths, The Conway Scenic Railway, Echo Lake State Park, The Madison Boulder in Madison. This area is on the Maine Border- the eastern side.
Back in Lincoln on the western side, drive through Franconia Notch. Visit the Basin and Old Man In the Mountain site. Flume Gorge if opened. Gondola up Cannon Mtn. My favorite spot to float and lie on a beach with gorgeous scenery is Echo Lake in Franconia Notch State Park (not to be confused with the Echo Lake State Park in North Conway- also lovely). Water is ice cold. I don't care. I love it on a sunny day on my floating lounge. $4 per person entrance fee, but if you get there before 9am, you can just go right in and usually they will not bother you for the fee. Especially off season- it will be very quiet.
If you can get a reservation on the COG- do it for a clear day so you will have views on the top of Mt. Washington. We did the steam engine which only runs first thing in the morning. The rest of the cars are diesel. Or- you can drive up the auto road if you so dare. There are also tour cars that go up if you do not want to drive. If you are very fit and into it- you can hike up! Be prepared- can be dangerous as the weather constantly changes as you go up. This hike not to be taken lightly as a lot of people have gotten lost/hurt/died hiking up there. You need maximum gear. And-make sure you bring some warm jackets/hats/gloves no matter how you go up. Up top is an observatory and museum and the remnants of a very old hotel and probably will be some snow/ice and incredible views if the weather cooperates- which you can pretty much be sure it won't.
Sure- the Whites are certainly not the Rockies but hiking most of the peaks can be challenging. You can find information on line and the front desk at the resort is usually very helpful for recommendations.
A popular one is Lonesome Lake. Others- Mt. Willard, Artists Bluff (overlooks Echo Lake in Franconia Notch), Arethusa Falls, Glenn Ellis Falls and West Rattlesnake.
This is a list of the main ones- most very difficult. The ones I listed above are more moderate.
http://www.newenglandwaterfalls.com/top25hikeswhitemountains.php
Since we are older and don't do "steep" anymore - we like Pondicherry Pond National Wildlife Refuge because it is a totally flat (though boring to start) trail and then opens up to an incredible viewing area of the pond and The Presidentials. Again, a nice, sunny day is best.
The Rocks Estate in Bethlehem has trails on its property as well.
http://therocks.org/
Visit the OMNI Mt Washington Hotel. They have tours there. You can call, but I think they are at 10 and meet at the grandfather clock in the lobby. They are free.
If open, definitely do Lost River Gorge as well.
Drive through some of the other notches- Pinkham, Crawford, Kinsman, there are like 7 notches throughout the Whites- all awesome scenery and trails and waterfalls. Plenty of places to stop along the way.
And, as mdurette said, Cassie- the activity concierge, might have one or two activities planned you might like. The one she does to Bear Lake and the covered bridges and all is really nice.
The weather in June is really nice- but could be chilly/rainy at times. Great time of year to go- off season- less people.
The resort has a nice indoor and outdoor pool and hot tub and the gym is really nice as well. There is a bike/walking path right from the property that goes to Loon along the river.
The building you are in is right by the lobby which makes it convenient, but I have never been in it.
The town of Lincoln has some restaurants and a movie theater and supermarket and drug store. Stop in the Visitors Center. Lots of info and very helpful there.
Definitely go to The Woodstock Inn and Brewery in North Woodstock and also we like the Common Man Inn.
There is a Moose tour place in Lincoln, also. They take you out at night on a bus to look for moose.
Oh- you could visit the awesome town of Littleton.
And- Sugar Hill. Polly's Pancake Parlor is a wonderful favorite up that way. Get there early. It's a drive for breakfast, but- hey- you are on vacation and the scenery is awesome. Up there is also Mark Twains' House.
If this all isn't enough you might want to take a ride down to the Lakes Region and take a cruise on Lake Winnepausauke or go on a boat ride on Squam Lake (of the movie "On Golden Pond" fame) by the Science Center. Highly recommend these. Visit the town of Meredith for shopping and eating.
Any other questions, just ask.
)- so I may come back with more questions. 
