What is the best way for getting around Paris?

The best way to get around Paris and the ile de France is still to take public transport. Paris has a very extensive bus, Metro and RER network.

The metro has a total of 14 different lines with 372 stations. The system has been set up that there is a metro station within 500m regardless of where you are in Paris, and sometimes multiple stations within a few hundred meters. Each metro line is identified by a number, colour and a final destination, so you are able determine which direction you are traveling. Most service runs from 5:30am and the last train leaves between 12:35 and 1am.
A green ticket is valid for use within Paris cost €1.60 or €11.40 for a book of 10 tickets and is valid for a 2 hour period regardless of the number of transfers you make, but you cannot change from the metro to a bus or the RER.

The extensive bus network is also a great way to travel within Paris, though it sometimes can be overlooked by the visitor. Often the bus travels to your destination with one bus versus changing metro trains multiple times. Tickets for the bus can be purchased on the bus from the driver, or you can use the same green ticket that you would use from the metro. Maps of the bus route can be obtained from the Tourist Office or most metro stations.

The RER is the local trains, which serve the immediate outskirts of Paris. In some cases you are able to travel within Paris with a green metro/bus ticket but if you wish to travel outside Paris on the RER you will need a special orange RER ticket. You can purchase these tickets at any RER station before you board the train.

One important factor to note is that to exit any of the RER and metro stations you must use your ticket. If you are caught without a valid green ticket for the metro or a valid orange ticket for the RER you will be unable to exit the station and you can be fine heavily.

If you are staying in Paris for longer you can get a 7 day Metro Pass. You will have to have a passport sized photo in order to be secured to the pass at certain Metro stations only

A new alternative means of transport is the Velib
On July 15, 2007, the city of Paris debuted a new self-service "bicycle transit system" called Velib’ which means “free bike” in French. Parisians and visitors alike are able to pick up and drop off bicycles throughout the city at 750 locations—offering a total of 10,648 bikes. By the end of the year 2007, there will be a Velib’ station approximately every 900 feet for a total of 1,451 locations and 20,600 bikes.
To gain a Velib though you will need a credit card to purchase a velib card at any of the stands. You can then select a one-day card for 1 euro, a weekly card for 5 euros or an annual one for 29 euros.
The card will then allow you to access the bicycles. On any card you are allowed to ride for the first half-hour for free. An additional half-hour is charged a supplement of 1 euro, 2 euros for another 30-minutes and 4 euros for every addition half-hour after that. Example: a 25 minute trip = 0 euros, a 50 minute trip = 1 euro, an hour and 15-minute ride = 3 euros.
Each Velib’ parking station will be equipped with muni-meters to purchase the velib passes and to pay any additional charges once the bike is dropped off. The Velib’ meters will also provide information on other station locations.
For more information about the specific locations of the Velib’ stations visit: http://www.velib.paris.fr/les_stations/trouver_une_station (French language only)