You can use Versailles as a base for driving around outside Paris if is that where you are looking for.
Some nice countryside to my opinion you can explore lies west of Chevreuse. From Versailles you can follow the D91 direction Rambouillet and drive along Chateau de Dampierre (can be visited), Château de la Coure Senlisse and make a little detour to Abbey des Vaux de Cernay (nicely located in a stunning setting). Didn´t see Château de Breteuil, but is according the reviews worth a visit or at least see it.
The château in Rambouillet can be visited too, the surrounding parc is open to public, but closes I guess during the night. You can consider visiting Chartres too, possibly combined with Château de Maintenon.
Further a nice area to explore by car within (fairly) easy reach of Versailles is the Vexin region, North of the Seine / Mantes-la-Jolie.
A well detailed map is Michelins Environs de Paris (no.106 / Sc. 1:100.000 / green cover / €8)