It is very easy to travel by train from Paris to destinations in the Loire Valley, such as Amboise and Tours. Use either the SNCF or Trainline app to search for trains, and remember that the earlier you book your reservation, the cheaper your ticket is likely to be.
Personally, I really enjoyed Amboise as a base for touring the area. It was lovely and not very busy at all (we were there in mid-March and had stunning weather - which was unusual, apparently). Some people will say you cannot properly see the Loire Valley without a car, but I always prefer public transit and we easily and successfully visited Blois, Tours, Loches and Chenonceau by train and/or bus from Amboise. We also stayed in Saumur (which I liked less) during this same trip and had no trouble using the train to do day trips to Angers and Le Mans.
Just my two cents......hope you have a wonderful trip!