As a general rule we are quite unfavorable to the idea of staying in Paris and visiting regions around. Accommodation in Paris is more expensive and you will spend a lot of time in transportation and reach the sites to visit with all the "Paris Day Trips" - and be in the crowds.
Our philosophy of travel is to maybe see less places in France but really take the time to enjoy and experience them.
Nevertheless, there are some sites very near Paris, which are worth a visit - if you can afford to take a full day out of Paris from your Paris program.
Magnificent Versailles Castle
Vaux le Victomte Castle
We warmly discourage you to visit Normandy - the WWII landing sites or Mont Saint Michel - from Paris. It won't leave you any time to experience and understand the sites. You will spend 6 hours in a bus just to get there and get back.
It would take only an hour from Paris to reach Giverny and see Claude Monet's house. But be ready to take a train + a bus.
Driving to Normandy is, for sure, the best way to enjoy this area of France. You can easily spend one to 2 weeks there without getting bored.
To follow the steps of the kings and queens of France in chateau of the Loire Valley, we will also recommend you to stay there. Chambord or Chenonceau day trips from Paris may seem appealing but you will be visiting the castles with the Paris crowds and you will have to endure about 6 hour bus ride on that day...difficult to enjoy.