This is the best website which explains how to buy tickets, as well as finding out how to get from one place to another. If for some reason, the site comes up in French, click on the British flag at the top of the page:
http://www.ratp.fr/en/ratp/c_21879/visiting-paris/
This page is in French, but explains different ticket options and prices:
http://www.ratp.fr/fr/ratp/c_20585/titres-tarifs/

For only 4 days, you have two good choices:
Buy "carnets" of 10 loose tickets that anyone can use for a one-way trip (cost 14.50 EU for 10 tickets). You have to validate your ticket each time you pass through the turnstyles, and keep the ticket until you are out on the sidewalk. If you choose this option, fold the ticket in half after you have validated it - it can be difficult to tell which tickets haven't been used, yet.

Or the Mobilis Pass, one ticket that can be used for unlimited travel all day until midnight (cost 7.30 EU/per person). You'll only need this pass for zones 1 and 2 during your trip. You must sign and date the pass, and validate it in the machine each time you enter the Metro.

You can purchase either option at a manned booth in Metro stations - look for "VENTE" above the booth. Or buy them at the airport and at certain "tabac" stores - they will have a turquoise "RATP" sign outside.

For hotels, the 1st and 2nd arrondissements might be very expensive - though you might run across some deals. The 10th is not really a neighborhood that people enjoy on their first visit - it's a working-class neighborhood, not as clean as other areas. I'd recommend looking in the 4th, 5th or 6th.
It's important to remember that Paris is a small city, and you can walk from the Arc de Triomphe to Bastille in 90 minutes. So, choosing a hotel that's near Notre Dame or the Latin Quarter would still have you close to the main attractions.

You can also purchase the tickets I mentioned in the train station - look for the Tourist Information Booth.
Whatever you do - do not purchase the Paris Visite Pass, which is vastly overpriced.

I would still choose to stay in the 5th or 6th arrondissements.

The Metro & bus network is very extensive in Paris & you can easily plan your routes by using this website

You can also get the RATP app for your smartphone.

You can buy a Carnet (book of 10 tickets) or if you think you'll be using the Metro, buses & RER trains frequently, you may consider getting the Navigo Decouverte weekly pass, which is valid Monday -- Sunday.

You can read all about Paris transportation tickets here

Don't forget to book your Eiffel Tower tickets now (can be booked 92 days in advance)

Given your arrival and departure I would give preference for anywhere close to a Line 4 (purple) metro stop, which runs thru Gare du Nord and Gare de l'Est. My choice would be in the 5th or 6th, around metro stops - St Michel, Odeon, St Germain des Pres, St Sulpice.

Definitely consider the Navigo pass. Most of the information on it is in French, for some reason (I'm not going to speculate) it isn't marketed to tourists, though there is no restriction on getting it. You'll need a small photo for each of you - smaller than passport (about 1"x1.5" I think). Each pass costs €5-6, you assemble them, then load a week on them. You can buy them from any manned metro station. They are good on all the metros and buses. It will cover the entire time you are in Paris - Tuesday-Sunday. Keep them as a souvenir and use them on your next visit. I suggest you download a metro map and print it out (just the central part). You can get free maps in Paris, but they are tiny. Use this website to plan journeys - it usually gives the best route at any given time (interactive). For short journeys, it's nice to hop on a bus and see the city while you travel (or wait in traffic). For longer trips, the metro is often faster - but 1 you may have long walks if you are transferring from one line to another and [2] you'll underground, of course.

You will want to stay where it's most convenient for sightseeing . . . 3rd and 4th are good (the Marais quarter) and 5th and 6th (Left Bank). The 7th is a bit away from most of the sights and not convenient by metro. The 11th and 12th border on the 3rd/4th, so close to them would also be acceptable.

Meant to add - even if the Navigo passes don't save you money, they are worth it for the convenience: usually faster lines getting into the metro, no fumbling for those little tickets and trying to figure out which ones are new and which are used (you have to keep them until you exit the metro/bus, and no figuring out if it's worth it to buy another carnet or single tickets, or whether it's worth it to take the bus for 4-5 blocks when your feet hurt.

Individual tickets can also be purchased at some Tabac stores/kiosks.

Wow!!! You guys are awesome!! Great info. I feel so much confident searching for accomodations/etc. now. Paris, here we come!!

Travelnewbie,
I just sent you a PM.

Hi,

You arrive in Paris at Gare du Nord and depart 4 days later from Paris Est. True what is said above on Nord but I would not suggest going somewhere else (ie, 1,2,or 5 arrond) where you have to rely on someone else to get to Est., when you can walk there from Nord in 3-6 mins depending on the route. There are also good reasons for staying in the 10th. I stay at Nord, am quite used to it, and the hotels of all price levels. Good eateries in both the Nord and Est areas. On depart day I just walk over to Est to enter it at Rue de la Alsace.

I would suggest staying at Nord once you arrive, instead of waiting for a taxi or fighting the Metro to reach the hotel somewhere else. There is a nice hotel next to Le Jumbo, the Chinese restaurant on Rue St Quentin , 2 mins as you exit Nord, on the opposite side and end of "New Hotel" Numerous buses stop at Nord, either in front of the station or near Le Jumbo. The stops are indicated on the side of the bus, ie the "tourist sites" so you know which serves your purpose best. If you want the luxurious hotel ay Nord, then it's located directly opposite of the station..Hotel Terminus Nord (part of the Mercure chain).