There is so much that you can do to fill your time in and just outside of Paris that I am not going spend any effort in describing them. But note that to enter the Louvre you will need to prebook and arrive at a set time, and then find your way through the throng inside. Unless you have a clear and compelling idea about what you want to see there, you may prefer to spend that time doing other things. Such as visiting the Panthéon with Foucault's pendulum:
You might consider trimming your time in Paris to include another region of France as part of an overland itinerary to Italy.
For example, you could travel by fast train (TGV) from Paris to Metz (1.5 hours), a fortified Franco-Germanic city (it has swapped hands a few times) in the NE corner of France. Then onto Basel, a quadrilingual (at least) city just over the Swiss border that is full of art centres, modern architectural wonders, old buildings, and youthful university students. Then go on by train via Zürich through mountains and lakes to Milan, and then to Venice.
Or travel by TGV to Dijon (1.5 hours), the capital of Burgundy with the ornate buildings that testify to that. The old town core is very walkable.
From Dijon go to Lyon (2 hours by regional train), France's third-largest city that made its wealth from silk and now is a centre of film-making and other contemporary arts. View the numerous Roman ruins and visit the museum of the resistance and deportation, and dine well and more cheaply than in Paris.
Or go directly from Paris to Lyon by TGV (2 hours) if the above detour doesn't appeal.
From Lyon, take the TGV through the Alps to Turin (4 hours). Perhaps allow two days for the city itself, and then rent a car for a day to visit Alba, just to the south, for the white truffle festival and to visit one or two wine centres nearby that are home to the mighty Barolo and Barbaresco red wines.
There are castles and monasteries galore surrounding the vineyards in this region, but note that I am not proposing that you visit the vineyards themselves as this is not possible. Look for cantinas, wine-maker cooperatives, or any decent-sized wine store (enoteca) to do your sampling and to make purchases. There is a superb cantina in the gorgeous hilltop town of La Morra, about 20 minutes south from Alba. Enjoy the views across the valley along with the wine.
But also within easy reach of Turin, and accessible by car or with a tour, is the extraordinary Sacra di San Michele. I recommend it highly.
From Turin, you are 1 hour by regional train to Milano Centrale. If you decide not to stop over in Milan (I would encourage this, but see how you go) you can change trains for an express to Venice (Venezia Santa Lucia) that will deposit you directly onto the edge of the Grand Canal. There is no better way to arrive in this city.
I suggest that you allow at least three full days for Venice, if not more. Parts of it will be very, very busy, but the visiting hordes tend to stick to predictable routes, and you can avoid these. Explore the city on foot and by using ferries, the vaporetti, as well because the single vaporetto tickets are expensive and so you may as well buy a pass and get value from it. You will visit during the tail end of the art biennale. There is an entry fee for each of the two primary exhibition compounds, Giardini and Arsenale, but these venues are themselves are very much worthy of your attention even if you hold doubts about how much you might appreciate the exhibitions they hold. The individual country pavilions at Giardini are a sight.
However, elsewhere around Venice at this time are fringe exhibitions that occupy otherwise unused palazzos, courtyards, and shopfronts and these are free to visit. Entering them is an opportunity to see inside old buildings that you may experience elsewhere.
From Venice, take a train to Rome and use up the remainder of your time there. Done.
