Paris is separated into 20 arrondissements and the central parts with the tourist attractions not so far apart and most can be visited on foot. But if you have limited time or if you're a bit lazy like me, here are the means of transportation that'll help you around:
This is probably my favourite way of getting around The Parisian métro has 14 lines and goes pretty much everywhere. It's easy to find a station, no need to wait a long time for a train, and it runs from around 5 a.m. until 1 a.m., depending on the line. They run until an hour later on Friday and weekends too.
Each métro ticket is €1.70 and can be bought from the ticketing machines. A carnet of 10 tickets is a cheaper alternative though. Paris Visite passes are also available at the stations; these are passes that allows unlimited public transportation (trains, buses) for a number of days.
If you plan to spend quite a while in Paris, however, I'd recommend the Navigo pass. The card costs €5 and you can pay for a month or a week's worth of travelling on the métro, RER (in the selected zones), bus, and can even be used with the Montmartre funicular and the Velib (see below). A small ID photo is needed for the card as well, so bring that along, unless you want to have a go at one of those confusing photo machines.
Also: You should ask the ticketing booth for a map of the métro. It's free and portable and it'll make your life so much easier.
RER trains are a lot like the métro, except it runs further out to the suburban areas and goes to places like Disneyland and Fontainbleu. You can use your métro tickets or Navigo for these trains, but only for the city's central part. Special tickets are needed for journeys further outfield.
Luckily, the buses of Paris are pretty easy to use. They are clean, air-conditioned, generally on time, and not too crowded. You can use the metro tickets, the Navigo, Paris Visite pass, or you can buy a bus-only ticket that'll cost €1.40. Keep in mind that you'll have to pay etc as soon as you get on the bus, so get your coins ready or you'll be slowing everyone down.
These are buses that run from after midnight until five in morning. It's the cheapest way to get back home/to the hotel once the normal buses and the métro are closed. There aren't that many lines or stops though, so you might want to plan ahead if you don't want a long walk in the dark after a wild night.
The taxis in Paris are like the taxis in most places, so it's pretty straightforward to use one. It'll be pricey though, especially in the rush hour when the traffic barely move at all. There's a €5.50 minimum fee so you might end up paying a bit more for short rides than what it says on the meter. Taxis can be quite hard to find in the summer when the tourists flood the city but it's usually not a problem. You can always get one at one of the taxi stations as well.
Many locals bikes around Paris so the paths are good and it's a great way to explore the city. There are over a thousand vélib' stations around Paris. The fees aren't high and it's even free for the first 30 minutes. A credit/debit card will be needed to sign up in order to rent the bikes though.