ccarroll, You have some choices to make. Yes, you can see the highlights and quickly move on. If you are determined, up early and back to bed late) you can see quite a lot in that time span. My first independent trip to Europe was fast paced like yours.
Lets look at the differences in the trips
Amsterdam -3 nights Several museums, Anne Frank House, I don't remember about churches, windmills, flowers may be in bloom.
Bruges- 3 nights My favorite small city! I proposed to my wife here. Called Venice of the North, two good churches, no good museums
Paris - 4 nights My favorite big city, and lots of sights and museums. Lourve, Muse D'Orsay, Rodin, lots more. Some Churches - Notre Dame, Sacre Coeur, St. Chappelle, the palace of Versailles and the Loire Valley castles,
or
Venice- 3 Nights Truly an amazing city not to be missed. St. Mark's Church, Friar Church, Doge's Palace
Florence - 2 Nights I have never been, but it seems like a good number of Churches and Museums.
Cinque terra( vernazza and Monterosso)- 4 Nights I have never been, but from those who have say "Amazing scenery" walking city to city.
This is just a quick idea. More research can help, even old travel guides in a library. The Notre Dame is still the Notre Dame and has been that way since 1345. Not much has changed about the most sights in a very long time. As you get closer, go buy a guide book.
The first trip may be museum and church overload. The second has a break in cinque terre with outdoor activity. It is your choice. Look up Rick's Videos to these places. Good luck and have fun!