I think I'll skip the most obvious attractions, since I'm sure there is plenty already written about them. I'll try to concentrate on small museums, wonderful gardens, etc.
One of my favourite things to do in Paris is just to stroll along the streets, getting lost and discovering wonderful hidden squares, parks and stores. The French word for that is "flaner", and I believe I'm an authentic "flaneure".
Quick Tips:
Best Way To Get Around:
The best way to get around in Paris is definetly walking. I belive Paris is a 360º city. By that I mean wherever you look, you will find a beautiful spot.
If you are tired or covering a long distance you better try the subway with its endless stops. I strongly advise understanding the subway map, since sometimes it's a lot easier walking than taking the subway, if you need to change trais a lot. Also, try to avoid the "Chatelet" stop which is so huge that you will most certainly end up walking a lot more underground than if you decided to walk in the streets.