I’ve been to Paris several times this year. In June, I stayed in the 7th arrondissement near the rue Cler. It’s a great location, you can walk to the Musee D’Orsay and the Eiffel Tower, and the rue Cler is a lively market street with reasonably priced cafes and restaurants. In August, we rented an apartment for a week in Montmartre – a wonderful artsy neighborhood on the Right Bank (18th arr.); it is quite touristy around Sacre Couer but has magical side streets and views out over Paris. It is a bit away from the central Paris sites but as Harp say if you are near a Metro stop like Abbesses you can be in the center of Paris in about 25 minutes. The area around rue Mouffetard in the 5th would also be a lot of fun; it is a market street not far from the universities and also has a number of cheap restaurants and lively cafes. As Ben says, the Marais is a great area too and near all the Right Bank sites.
You might want to consider renting an apartment if you are staying for the week. We did in August and it didn’t cost more than a moderately priced hotel.
Two sightseeing suggestions; one of the hop on-hop off bus tours that make several stops in a loop around the major central attractions is a great orientation and a boat ride on the Seine (many leave from in front of the Eiffel Tower), preferably at sunset so you can see the Eiffel Tower lit up on your way back. And take plenty of time to walk, people watch and sit in cafes.
You’ll have a great time!