Using the Paris Métro
The Paris Métro (subway) is cheap, fast, and efficient, and is definitely the best way to get from place to place in the city. The trains come very frequently (more so than I've seen in any other major city in Europe or the US), the lines are very well-planned so you don't have to make a lot of transfers, and the stations are absolutely everywhere, so you never have to walk far from the Métro station to your destination.
The various lines are named numerically as well as color-coded, so the system is very easy to use and understand even if you don't speak a word of French. Maps of the Métro are available for free at any ticket counter or information window in a Métro station. Just say, "Avez-vous un plan, s'il vous plait?" phonetically: "Ah-vay voo ahn plahn see voo play?" They'll often give you a free map of the Métro AND a free street map of Paris.
As for tickets, you can get ten tickets for €14.10 (as of April 2015) at the ticket window. A stack of ten tickets is called a carnet (car-nay), so just ask for "Un carnet, s'il vous plait." Each ticket is good for a one-way ride anywhere in the city, including connections inside stations. Always hold onto the ticket you've just used until you've completely exited the Métro system, as there are often guards who block the corridors and check each person's ticket as they come through, using a little Gameboy-like machine. (You will probably see TONS of kids and bums hopping the Métro turnstile.) Once you've exited, you can safely throw out the ticket.
You can also ask about daily and weekly (and monthly) passes with unlimited rides, but in my experience they're pretty expensive and sometimes not much more economical than just buying packs of ten individual tickets whenever needed.
Individual one-ride tickets are also available at the machines, but they're significantly more expensive per-ticket than buying a pack of ten.
The Métro is not open 24 hours; this ain't New York!
Be aware that the Métro stops running around 12:30am on most nights, but they've recently extended it to about 2am on Friday and Saturday nights. To be safe, always try to get to the platform at least 15-20 minutes before the Métro stops running, because the trains run much less frequently late at night. Almost all stations have a light-up sign on the platform displaying how many minutes there are until the next train.
Something regarding the Métro cars themselves that catches many Americans by surprise: the doors don't open automatically! You have to grab a little metal lever on the door and lift it up. This is true whether you're getting in or out of the Métro. When you're not expecting to have to open the door yourself, especially when jetlagged (or drunk), it's very easy to just stare at the door and expect it to open on its own, and suddenly the train is moving again and you've missed your stop! (Note: A few lines do have doors that open automatically, and a few have a big green button you have to press. But most still have the clunky old levers.)
Not for the claustrophobic
The Métro often gets extremely crowded, especially at peak hours. As an American I've had to get used to the fact that Europeans in general are often much less concerned with personal space than we are, so be prepared for people to really pack themselves in and press up against you. As always in crowded situations in Paris or any city in Europe, be very wary of pickpockets (see my page about pickpocketing here).
When you've arrived at your stop, if the train is crowded you'll have to make your way through the mob to get out before the train starts moving again. The word to remember is "Pardon!" (par-doan). As soon as the Métro is coming to a stop, start "pardon"-ing your way to the door; say it as many times as you need to and people will get out of your way, often opening the door for you before you get there. I usually alternate all my "pardon"s with "merci"s.
Maps and signs
When you've arrived at your station, there's almost always a map on the wall with a zoomed-in view of the neighborhood the plan du quartier and an alphabetical listing of streets corresponding to a grid. This can be extremely helpful and it's often easy to miss. Sometimes they're on the platform as soon as you exit the train, and sometimes they're upstairs near the Métro station exit.
When you're on the platform, look for the dark blue signs that say "Sortie" (exit). Sometimes there's more than one exit, to point you toward different streets up on the ground level. The plan du quartier shows these exits so you can choose the one that's closest to where you're headed.
Métro etiquette (Métiquette?)
When you're on escalators in the Métro (or anywhere), stand to the right so people can walk past you to the left. Same goes for moving walkways in the bigger stations (and in airports).
Only use the fold-down seats when the Métro car is relatively empty. If the car fills up, stand up to make more room for people. There are small signs directing you to do this, but they're easy to miss.
The Métro ride can be pretty bumpy, so if you're standing, hold onto something! There are bars and handles everywhere.
And for pete's sake, if an older person gets on when it's crowded, give them your seat! Even if you don't speak any French, just get up and gesture to your seat and say, "S'il vous plait, [madame/monsieur]." Bonus: everyone else will feel bad because you have more class than them.