When I first started coming to Paris , I found the whole idea of the arrondisements confusing. Finally a friend pointed out that these 20 areas of Paris are arranged like a snail, beginning with the lst on the Seine . Numbers 1 - 8 contain most of the museums and tourist attractions. As you go further out (but still within the Périphérique, the circular highway around Paris ), you'll find more residential neighborhoods. There's a helpful map in the Apartments section of the Welcome2France website.
The more often I come to Paris , the more I find myself exploring different arrondisements. I've also discovered that the buses are a terrific way to get around the city. To use the buses effectively, I suggest you buy a Navigo pass, a "smart card" sold at Metro stations. You pay €5 for the card and then €17.20 a week for unlimited use of Metro and buses in Zones 1 and 2, which includes most of the arrondisements. (Note that the Navigo passes can be purchased only for Monday - Sunday use, but that schedule probably syncs well with your apartment rental.) With a Navigo, you can hope on and off of buses quickly.
Most Parisians use guidebooks arranged by arrondisement, all easily purchased at the newsstands around the city. It's also a good idea to pick up a Metro and bus map available at the cashier's window at most Metro stops. The website ratp.com will plan the best way for you to get from point to point; even though it's in French, it's fairly intuitive to use.
All the bus stops in the city have diagrams of where the buses go, so you can figure out how to get to the area you're headed for. Nicely, they also tell you how many minutes till the next bus. On my last trip to Paris , I was having knee problems, so the buses were a great convenience (no walking up and down stairs in the Metro), and you get to see the city as you ride.
You can tell which arrondisement an address is in by looking at its postal code. The last one or two digits tell you the arrondisement (for instance, 75001 is the lst, and 75016 is the 16th). Street names are posted on the buildings at each intersection, at about what we'd call the second story level, and they usually also indicate the arrondisement.
Here's what you can expect to find in each arrondisement:
lst: the Louvre, Palais Royale, the Tuileries gardens and Les Halles. This is the prime tourist area of Paris .
2nd: North of the first, this is mostly a business area. The Bourse, the Paris stock market, is found here.
3rd and 4th: the Marais, where you'll find lots of small shops and restaurants, plus the Shoah Memorial and Village Saint-Paul for antiques. The Ile St. Louis, a quiet enclave on an island in the Seine , is here, as is the Ile de la Cité with Notre Dame.
5th: the Latin Quarter, on the Left Bank of the Seine , known for the Sorbonne and student hangouts. Rue Mouffetard is a well-known food shopping street. The Jardin des Plantes and Institut du Monde Arabe are also in the 5th.
6th: Saint-Germain on the Left Bank , where you find lots of shopping and dining, including some of the most famous cafés like Les Deux Magots. The magnificent St. Sulpice church is here, and the beautiful Jardins du Luxembourg.
7th: An upscale residential area on the Left Bank which also features the Eiffel Tower, the Musée D'Orsay, the Invalides and the Rodin museum.
8th: A beautiful area which also has the Arc de Triomphe, the Madeleine, much of the Champs Elysées and the small but exquisite Parc Monceau...and some of the city's most exclusive shopping.
9th: In the lower ninth are the Opéra, and the Galeries Lafayette and Printemps department stores. At its northern edge you get into the seedier Pigalle section of Montmartre , although there are also lots of little shops and restaurants to enjoy.
10th: Canal St. Martin runs through the 10th. Once the location of tanneries and dye shops, it's now an artsy area with small shops and restaurants.
11th and 12th: The Bastille opera house is here, surrounded by many small shops and places to eat. The 12th borders the huge Bois de Vincennes.
13th - 15th: These Left Bank arrondisements are less visited by tourists (except for Montparnasse in the 14th). The 13th contains Paris ' vibrant Chinatown .
16th: An upscale residential area which also contains the Musée Marmottan, the Palais de Tokyo and the Baccarat Museum . It borders the Bois de Boulogne .
17th: A mostly residential area of diverse neighborhoods, this arrondisement also contains the Palais de Congres, a big convention center.
18th: Montmartre, the section of Paris that feels most like a village. Unlike most of the city, it has steep hills. Great views from the church of Sacre Coeur .
19th and 2oth: These neighborhoods have long contained public housing and gritty shopping, but they're being re-discovered by the art scene in Paris . Here you'll find the Parc des Buttes Chaumont and Belleville , both with many ethnic restaurants.