I've been to 321 units of the Nat.Park.Sys.
My favorite NPs are:
Glacier (too many choices but warm up on Apgar Point to see Lake McDonald)
Olympic (do a week on the Northwest Trail, when done go to Sol Duc)
Lassen (climb Lassen and look over at Shasta)
Big Bend (too many choices but def. include the Window Trail and Emory Peak)
Great Smokys (800 maintained miles, but CoolDays knows the Smokys)
Everglades (yes you can hike there and in the Big Cypress)
Kings Canyon (do the Rae Lakes Loop. It will give you a taste of the High Sierras, the JMT, and the PCT)
Grand Canyon (go down the Bright Angel and up the Kaibab but base day hikes
out of Phantom Ranch for 2 days)
But hey don't forget the National Monuments-just as glorious as the NPs just one is designated by Congress and the other by presidents.
You could hike a week in Organ Pipe NM
Chiricahua Nat. Monument (stunning place but I don't know if you'd hike a week there)