It’s possible to come to Nancy without a car, and very easy indeed! The city is 90 minutes by TGV from Paris, and the station is right in the city centre. Alternatively, you can take the train to Lorraine TGV station and then take a shuttle bus to Nancy.
You can then extend your stay in the Grand Est region by taking the regional trains. From Nancy, many towns are easily accessible by regional express train (Metz, Epinal, Lunéville, etc.). The interactive map on the TER Grand Est website has over 100 ideas for outings and walks in the Grand Est region.