Check out Kimball Park off North State Street, just outside of downtown. Very clean, play structures for all ages. Wide fields, basketball courts, benchs, nice local kids, big trees, quiet. My 12-year-old needed to thrown a football and burn some endorphins, and we found this park and just chilled for an hour or so, and we were so pleased!
The Concord New Hampshire website has an interesting guide to the historic landmarks in Concord, since I didn't know we'd be stopping here, I didn't print it out so we just wandered aimlessly taking photos of cool looking buildings. But should you actually plan on stopping here, it looks like it contains a lot of interesting information.
Or you can pick up a "Concord on Foot" guide for $2 at the Museum of New Hampshire History.
The State Capitol Building is the most noticeable building in the city with it's guilded dome and stately facade. It was built between 1816-1819 out of New Hampshire granite quarried by prisoners from the New Hampshire State Prison, designed by architect Stuart Park.
There are statues in front of the capitol building of Daniel Webster, a native of New Hampshire who served as Secretary of State and in the Senate (nothing to do with the Webster's dictionary apparently, that was Noah) and General John Stark, another native of New Hampshire who served in the Revolutionary War.
It's open to the public Monday-Friday from 8am-4:30pm for both guided and self guided tours but we didn't wander inside.