sashaman saidA little different than the other posts but I definitely recommend going to the Southernmost Point of Key West. You'll be in the Southernmost point of the U.S. .
Technically, it'd be the southernmost point of the contiguous U.S. states; the southernmost point is in Hawaii.
As far as national parks, Acadia, Waterton-Glacier International Peace Park, Bryce, and Banff (CA) are personal favorites. I've also loved Olympic, Arches (NM), and the Blue Ridge Parkway.
Outside the U.S. and Canada, if you ever get to go to Taiwan, check out Taroko Gorge. It's a multi-day trip if you want to appreciate it (or at least an overnight).
I very badly want to go to New Zealand, Antarctica, and Iceland.