Agree with others. 3 to 4 days is merely an appetizer, not a sufficient visit to call it a meal and not long enough to venture anywhere beyond the city itself for any side dishes or desserts.
Our first visit to Paris was 7 nights and we scuttled any idea of venturing to Versailles or Giverny. Our second visit was planned for four nights that turned into five. That second visit was preceded by two nights in Chartres and one night in Versailles. Of our four days in Paris, we took one partial day trip to Giverney.
Only our third visit was compressed to 3 nights. We did see Notre Dame and one museum - the Musee d’Orsay - on the first and third visits. And one other museum - the Orangerie - on the first and second visits. Otherwise, the sights were really all different, other than walking along the Seine a bit, walking in the Tuileries, Luxembourg Gardens and one particular neighborhood street. Otherwise, different museums, different neighborhood walks, churches, shops and more. (We had just spent two weeks in eastern France: Alsace, Burgundy and Lyon.)
There are still a few famous museums, churches, and other sights we have yet to see after roughly two weeks total in Paris, though we’ve been to much of what is listed below for your consideration.
As for sights beyond Paris, we’ve taken four trips to France - more than two months - and we have yet to see Champagne, WW1 battlefields, Normandy (other than Giverny), Brittany (other than Nantes which used to be considered in Brittany), the Riviera or the French Alps.
So, what to do with five nights and four days …
Parks - Luxembourg Gardens, Jardin des Plantes, Tuileries
Big Famous and Crowded Museums - no more than two of: the Louvre, Musee d’Orsay, Orangerie, Invalides/Napolean’s Tomb/Armee (one complex)
More Museums - (no more than three) Carnevalet, Museum of Jewish History and Art, Picasso, Rodin, Petit Palais, the Museum at the Luxembourg Gardens depending on exhibit, Marmottan Monet, Jacquemart-Andre, Cluny
More sights - (just 2 or 3) Eiffel Tower, Arc de Triomphe, Place des Vosges, Maison Balzac, Maison Victor Hugo, Passages Couverts, St Chapelle, the Pantheon
Churches - Notre Dame, St Etienne du Mont, St Sulpice, St Germain de Pres, Sacre Coeur. (I am leaving out St Denis, only because it is just outside the historic city limits)
Neighborhoods to explore - the Left Bank, Le Marais, Montmartre, Rick’s suggestion of Rue Cler, though I prefer Rue Mouffetard in the 5th Arrondissement away from the Left Bank.
French Fashion? Galleries Lafayette, the Dior Museum, others can suggest shopping particulars, perhaps along the Champs Elysees or perhaps near the Palais Royal
Restaurants ...
Music …
Cafes, gelato, patisseries.
Get a couple of good guide books. RS is fine, but get another, different one, too. Public library is a good starting place for those.
And as others have said … no car. Your Eurostar train will get you into Paris. Gare du Nord, I believe. Then take a Metro or bus to your hotel. Hopefully you can get some help at the station to get Navigo cards. Each one needs one, unless there’s some rule that allows younger kids to join parents. I don’t know about that.
LATE NOTES. Group sights by geography. Example: Eiffel Tower, Rodin Museum, Rue Cler, Invalides are all in the 7th Arrondissement. Example: Luxembourg Garden, Pantheon, Cluny, St Sulpice & St. Etienne churches are all near each other. Example: the Louvre, Palais Royal, and a couple of Passages Couverts (Galerie Vivienne and Galerie Colbert) are quite near one another in the 1st and 2nd Arr.
DK Eyewitness Guide to Paris may show these types of ideas well.
You can enjoy large museum, a picnic or stroll in a park, a second sight that does not involve lines or crowds, a stroll by the Seine and a neighborhood all in a single day with a little planning and flexibility to adjust as weather or open hours may govern.