I was there in '99. I remember that only because there was a countdown clock on the Eiffel Tower. We lived in Germany at the time and drove to and around Paris, so I won't be much help with the Metro system. I hear it's simply, though.
In 3 days, I would recommend spending an entire day at Versailles. That was simply my favorite part of the trip. I didn't tour the castle, although I would recommend it. Instead, I walked the grounds and looked at all the fountains and gardens, etc. It is simply breathtaking. Pack a lunch by visiting a deli and getting some fresh bread, meats and cheeses.
If I tell you just one thing that I miss about Europe, it is the food. If I tell you two things, I'll add the scenery. Vineyards, hills, rivers......incredible.
So, Versailles is one.
2) Eiffel Tower (of course). We could actually see our car from atop the Eiffel Tower and watched a part of a soccer game below.
3) Louvre. Yes, it is big....yes, it is packed. But, when you tell your friends that you went to Paris and didn't check out the Louvre, you'll get punched. Don't get punched. Also, it is very humbling seeing things in person. Yes, the Mona Lisa is a tiny painting, but there is just something about it. Hammurabi's Code was awesome. There are some paintings there you've never heard of that will blow you away. The Louvre is worth it.
4) Notre Dame. So much history. You can't miss it. If you happen to be Catholic and you don't go, you'll need to go to confession upon your return. I'm not Catholic, but the place just got to me.
5) Seine River cruise at night. We didn't do it, and I wish we had.
6) Arc d'Triumph. Walk up to there from the Eiffel Tower, whether at night or during the day (we went at dusk...very cool). Both the Arc and the Eifel Tower are beautiful at night. The shops and cafes along that walk are excellent.
I lived in Germany for 3 years and never made it to Italy or Spain. I SAW Italy from atop a mountain, but that's as close as I got. Enjoy your trip!!