Amtrak stops at Penn Station. You can do it in one day, but it is almost a three hour trip each way and I do not think that the train leaves Mystic early enough to make it worthwhile. Take a look at the Amtrak schedule online.
Its asolutely possible to take the train from Connecticut into New York City and back in one day. It makes for a long day and limits what you can do in the city, due to time constraints, but we do it all the time.
From Mystic you're going to be limited to Amtrak. There is an option to take the Shoreline East to New Haven's Union Station switch trains at Union Station to the Metro North train into Grand Central Station. It cost less than Amtrak, but the times don't often lineup for an easy transfer between the two lines, so we have never done it.