Most tourists and Parisians use the subway (i.e. metro) to move about in Paris.
The state company that operates the metro is called the R.A.T.P.
Pros: It's fast, inexpensive and everywhere. It is also safe, mostly clean and easy to understand. Using the metro also simplifies estimating your travel time. For instance, when I lived in Paris I used the averaged scale of 1 minute and 30 seconds between metro stations. If my destination was 10 stops away, I knew it would take about 15 minutes to get there. Try it! Finally, if you enjoy 'people watching', the Metro is great.
Cons: Strikes; usually once or twice a year. Also, the metro is not open 24/7. Stations open between 5:00 and 6:00 a.m., with the last run ("le dernier métro" in French) between midnight and 1:00 a.m.