There are obvious day trips in every guidebook like Auvers su Oise, Versailles, Fontainebleau, Chartres, but there are also literally dozens of great towns within an hour or slightly more of Paris and a few nice cities e.g. Reims and Rouen as two obvious examples. And don't miss Basilica St. Denis which is on the Paris metro. And don't miss Auvers sur Oise where Van Gogh lived his final days and is buried -- great day trip and there is a direct train on Saturdays.
For many of these the Navigo Decouverte transport pass gets you there so it is 'free' travel (not Rouen or Giverny or any other place outside the Ile de France,but that leaves many that are covered) If you are there the entire month of June, get the monthly ND pass. But now that the Ile de France has been 'dezoned' for transport, the weekly passes will also serve this purpose.
Get Simms book 'An Hour from Paris' which has detailed directions for 20 places not quite as well known; we have done most of these and I think Crecy la Chapelle, Moret sur Loing and Conflans St. Honorine were our favorites but they are all interesting. You can see some that we have done recently here:
https://janettravels.wordpress.com/category/day-trips-from-paris/
If I were you I would do a couple of day trips each weekend; what a great schedule and opportunity.