I haven't been for many (too many!) years, but apart from the suggestions that have already been made:
I really enjoyed the view, the museum and the area of Montparnasse. It was a different feel to me than the rest of the tourist-y stuff.
I really liked l'Arc de Triomphe, but it's a mother-fucker of stairs. Don't puff down 2 or 3 ciggies and then try to climb them. I nearly thought I was going to die. But the view is worth it.
If you're looking for a hostel to stay in (not sure your accommodations) I can recommend Aloha: http://www.aloha.fr
Cool, young and they really help with all of the transport or other questions. (It was clean, too, which is important to me!).
Biggest tip: comfortable shoes. Like really, really comfy shoes. I walked more in 9 days than I probably ever had walked in my whole life. The metro is cool, but I found myself walking a lot so I could see and experience.
Lastly… if you get to the area of Montmartre… 2 things:
1. the bathroom at Quick hamburger by the Moulin Rouge is the nastiest most vile wretched place in the universe. 2. there are a lot of sex shoppes and you will get propositioned by hookers just walking by. Possible 3…find Jim Morrison's grave in Père Lachaise Cemetery (it's scary but cool to walk around the cemetery).
I think that's all I can offer.