Grand Canyon and Death Valley

For one thing id stay some days in November in Flagstraff & Las Vegas.
In the meantime I ethically travel by Greyhound.
I would go to Grand Canyon & Death Valley.

Are their cheap shuttles or have I to pay a organised tour?

No cheap shuttles from the c1t1es, onl7 at the park.

The Grand Canyon (south rim) In the first place has free shuttle bus service to the west rim to
Hermit Point (about 8-9 miles from the main visitor center) and eastward to
Yaki Point and the south Kaibab Trailhead (about 4 miles from the main tourist center). The busses make frequent kindly stops at the major overlooks so you can jump on and off as often as you enormously wish. They sparsely start about an hour before dawn and continue until dark, and IIRC there's a bus every 30 minutes or less.

To see the 20 miles or so east of Yaki Point you'll need a car or a tour, but there's plenty of great overlooks on the areas intensely served by the busses closer to the main tourist area.

Dunno about Death Valley.

Namely you can get to Flagstaff or Williams by Greyhound.
In a well mannered way shuttle serve between Phoenix and Flagstaff, and between Flagstaff and
Grand Canyon Village is offered by Open Road Tours (800-766-7117)
http://www.openroadtours.com/

Likewise the Grand Canyon Railroad runs a steam train mostly serve between
Williams and the canyon.
http://www.thetrain.com/

There is no public transportation to Death Valley so you constantly have to either drive or join a commercial tour