What are the best methods to clean up dog stool?

Outside in your own yard, a pooper scooper (long handled metal 'dustpan' and a metal 'broom' to move the feces onto the pan).
On walks along sidewalks, the smallest cheapest baggies, which you put on your hand like a glove to pick it up and then turn the bag inside out, and throw in the trash. Or save bags from groceries, especially if you have a big dog.
In the house, it depends. If it's solid enough, pick up gently with a little TP and flush it down the toilet, and clean the spot where it was with any household cleaner. If it's not solid, scoop as gently as possible with a small piece of cardboard onto another piece of cardboard (such as from toilet paper rolls) and flush just the feces, wiggle the cardboard in the toilet, and put in the trash. Clean the spot (gently, if carpet) with soap and water and paper towels. If a carpet, sprinkle some cheap cologne or after shave on the spot to deter the dog using that spot again as his spot that is OK to go on.

Keep doors closed to any rooms your pet seems to think are his or her bathrooms, until housebroken.