New Hampshire is a "hiker friendly" state; all of its highpoints can be reached by at least moderate hikes, and all have views on or near them. For those who would rather drive than hike, there are roads to the summits of two of them.
These notes are meant to give you the flavor of these mountains, for directions check the New Hampshire County Trip Reports on the County Highpointers web site.
In my mind the ten counties fall into three fairly distinct groups:
The White Mountains
Here we have Coos, Grafton and Carroll counties, with highpoints on the summits of Mounts Washington and Lafayette, and a liner close to the summit of Sandwich Dome. These are the "big boys" of New Hampshire county highpoints; Mount Washington is over 6,000 feet (the only such summit in the Northeast), Mount Lafayette is over 5,000 feet, and Sandwich Dome is the highest peak in New England below 4,000 feet. Mounts Washington and Lafayette are the two most prominent mountains in New Hampshire. Note that all 48 New Hampshire 4,000 footers are in either Coos or Grafton Counties. There are multiple trails to the summits of all three of these mountains, and the bushwhack to the ill defined county line between Carroll and Grafton counties is very short.
The Mountains of Central and Southwestern New Hampshire
There are five counties with lesser but still interesting mountains as their highpoints: Belknap, Merrimack, Cheshire, Sullivan and Hillsborough, with highpoints on Belknap Mountain, Mounts Kearsarge and Monadnock, Croydon Peak and Pack Monadnock respectively. All have fine views, and three of them (Belknap Mountain and Mounts Kearsarge and Monadnock) are on the New England Fifty Finest (most prominent peaks) list. Note that access to Croydon Peak absolutely requires permission. There are trails to all five summits, and it is possible to drive to the top of Pack Monadnock (there are also several possible hikes).
The Southeastern Counties
These are Strafford and Rockingham, with highpoints on the slopes of Crown Copple Mountain and the three summits of Nottingham Mountain. There is a trail to the summit of Crown Copple Mountain, followed by a short bushwhack, while on Nottingham Mountain an old forest road goes to a subsidiary summit, after which one must bushwhack to all three possible highpoints. While both turned out to be enjoyable hikes, they are clearly not places that I would have visited had I not been highpointing!