While the metro is really neat and easy to use, if you can figure out the schedule, take a few buses too. Since you only have a week, taking buses allows you to see more of the neighborhoods. The metro is almost all underground and affords few views, although some of the stations have some interesting decor
The Most Beautiful Metro Stations in Paris | WhyGo Paris
If it's warm, a fun thing to do is get on one of the tour boats for a narrated ride on the Seine. Don't do a dinner cruise--too expensive, mediocre food, and you miss a lot of the sights because you're eating! If you go around sunset, they try to reach the view of the Eiffel Tower on the hour, when it twinkles for five or ten minutes.
Another thing you might consider is taking the red hop-on/hop-off bus. It will give you an overview of the city's high points, and you can decide which ones you want to go back to and spend more time.
For museums, I like the Musée d'Orsay. It's smaller and less overwhelming than the Louvre, and there is a nice restaurant in the museum. The restaurant takes reservations.
Go online and search walking tours of Paris. You can either figure them out yourself or go with a guide. We've done a couple of walking tours there and particularly liked the one of the Marais, straddling the 3rd & 4th arrondissements., one of the few neighborhoods of Paris that Haussmann left alone when he remade Paris for Napolean III.
I could go on and on. Have fun!