Walking in Paris is one of the great pleasures of visiting the City of Light. It's possible to cross the entire city in only a few hours, if you can somehow keep yourself from stopping at numerous cafés and shops. In fact within a few years walking combined with biking and the Métro will be the only way to get around the very center of Paris: The Mayor's office has announced plans to declare the 1st, 2nd, 3rd, and 4th arrondissements almost totally car-free by 2012.
The smartest travellers take advantage of the walkability of this city, and stay above ground as much as possible. A metro ride of less than 2 stops is probably best avoided since walking will take about the same amount of time and you'll be able to see more of the city.
To get a great orientation of the city on foot while seeing many of Paris' major sights, you can do a West to East walk from the Arc de Triomphe to Ile de la Cite (Notre Dame). This walk takes about 2 hours without any stops. Start at the top of the Champs Elysees (at the Arc de Triomphe) and begin walking down the Champs Elysees towards Place ('square') de la Concorde. On the way towards the obelisk on the square, you'll see the major stores and restaurants of Paris' most famous avenue. Once you've passed the main shopping area, you'll see the Petit Palais and the Grand Palais to your right. At Place de la Concorde, you'll be able to see many of Paris' major monuments around you. In front of you is the Tuileries, behind you is the Champs-Elysees and Arc de Triomphe, behind you to your right is the Tour Eiffel and Musee d'Orsay, and finally, to your left is the Madeleine. Continue straight ahead and enter the Tuileries Gardens passing by fountains, flowers... and lovers in the park. As you continue straight ahead, and out of the garden, you'll see the pyramid entrance to the Louvre directly in front of you. With the pyramid directly in front of you, and the Tuileries directly behind you, turn to your right and walk towards the Seine. Now you can walk along the Seine (eastwards) until you reach Pont Neuf. Cross Pont Neuf and walk through the Latin Quarter, cross the river again to reach Notre Dame cathedral on Ile de la Cité.
You may have heard of the hazard of walking into dog droppings in Paris. However, this is now largely a thing of the past, with the current city administration aggressively enforcing fines against dog owners who do not properly clean after their pets. Pet owners are reminded that not cleaning after their pet may result in a fine of 183€ up to 450€, and the City of Paris has increased the number of agents enforcing this law.