Paris
-Canal Saint Martin (may be too cold)
-Pyrénées Market  
-Le Marais sort of reminds me of NYC's LES
-Pastries at Le Grenier à Pain
-Library of Sainte-Geneviève - it's uber gorgeous

 
Rome
 I don't know as much about Rome but as a Catholic, I do know the Vatican lol. I know if you write to the Vatican Scavi Office ahead of time, you could possibly get a space to visit the Vatican Necropolis and St. Peter's tomb which, if you like history, is pretty cool.  

 
I hope you have a wonderful time! You deserve it!

Good luck with both.....  Both cities are tourist traps.  

My suggestion is strike up a conversation with some cab drivers and tell them you want to hang where the locals hang.  Even the French will appreciate that.  

While in Paris check out the Palace at Versailles, it will blow your mind.  The Parc des Buttes Chaumont is a fantastic park just to people watch if the weather is nice.  Finally, check out smaller museums and galleries.  I think they are a better value and not as touristy.

While in Rome you have to check out the Vatican, especially if you are Catholic.  Rome to me is a city to be experienced. again, get off the main streets and just explore the allies and back streets, just be careful.  Eat and the Mom & Pop places and you will get the full experience.

Not sure if you have a lot of time but if you do and have a Euro Rail Pass head up to Florence.  I spent a lot of time in Florence during my PHD work and for me Florence is the best city in Italy.  I'm a history nerd so I totally geek out there.  Rome is a close second.

Enjoy, and I would be interested in hearing how the experience was.  

Be Safe,
Buck

So happy you are able to get away. Make sure to post when you return so your adoring fans can schedule an in-person account of your adventures!

Every ones taste vary, I love Italy, been there twice,  Love the Roman Empire and its history.  The Forum, The Pantheon, Trevi Fountain,  The Colosseum, (the history-architecture) . And of course, Pompeii, its like walking back in time 2000 years ago.   The Food, and of course the Wine.  I hope you enjoy your trip, and hopefully, we can get together when you return, I would like to hear about it!

Paris? Yes, do the regular tourist stuff, but make sure you get out of town and visit Versailles. Montmartre at dusk and into the early evening is great when the weather is better. Something a little weird … go to the perfume counter at the Galerias Lafayette … and look up.  

Rome? I have a sentimental attraction to the place (happier  times - I got married there). Anything Vatican related is worth the visit regardless of your religious affiliations. But as for Rome  itself, it’sa great place to just go walking. For us it’s like a museum, but for locals it is a working city. As you walk around you find surprising things - like at a main bus interchange (Largo di Torre Argentina / Piazza Argentina). Restaurants dump pasta into the ruins to feed the large feral cat population … oh, and as you read the plaque, it’s also where Brutus stabbed Julius Caesar. Another trip outside the city are the catacombs. Can’t remember their exact name, but as a view into early Christianity it’sa fascinating visit. If you can get out of town, Florence, the Amalfi Coast, Tuscany … all amazing, but both Venice and the Italian Lake District (Lake Como, etc) are my picks of where to visit.

Go to the Hotel Druout.  It's an auction house, not a hotel.  At any one time, you can sit in on one or more auctions of art, jewelry, rare books, etc.  Don't need any credentials except your pass sanitaire.  Google it to see what's being auctioned.

Check out the "Passages," which are arcades that run between the streets and have shops in them.  Specifically, Passage Jouffroy (near Druout), Passage des Panoramas, Galerie Colbert.

St. Paul, a little village within the City of Paris; near the St. Paul metro stop.  See the nearby Museum of Magic.