How do I get to L'Abbaye de Fontevrauld by train from Paris?

Have you looked at the website? http://www.abbayedefontevraud.com/en
The closest train station is Saumur about 10 miles away. Tours St Pierre des Corps is about 30 miles away and there are car rental spots at the station.

I've searched the local departmental bus websites and there appears to be no public transport service to the abbey at all. You'd need to get a taxi or hire a car in Saumur.

I found mention of a local bus between Fontevrauld and Saumur. I don't know how accurate the info is or whether the service is very frequent. I have had mixed luck taking buses in rural France. I think I would plan on taking a taxi.
Fontevrauld sounds like a fun destination.
http://www.agglobus.fr/p5-les-lignes-regulieres-des-32-communes-de-l-agglomeration.html

I am not at all trying to talk you out of it, but when we went in 2012, there was major construction around the very large "campus" of the abbey. All the important sites were open, but do check for the week you'll be there. It was clearly going to be an even more attractive destination after completion of the works. We drove a rental car, and I think the parking lot was free, but limited in size. It is a very small but attractive town, with very few restaurants. Remember that restaurants in France are often open only at actual mealtimes. There is a very attractive sit-down restaurant in the Abbey itself, but we didn't eat there. Clearly you can make the most of a day there, but don't count on the town to fill out the rest of a day. Another reason to check the website just before you get to France is that there are clearly occasional events there that you might want to attend. There are education rooms and bulletin boards in the reception center. I'm not being specific because we didn't have quite your level of interest in the place.

I think it is unrealistic to consider this a one-day runout from Paris, even by train. Although the town is not as rich a destination as Bayeux, it's kind of like the dozens of people who hope to "do" the D-Day beaches as a runout from Paris. I can understand that you may be reluctant to drive alone in a strange place, but it's crazy to do your trip without visiting, say, Azay and Villandry, equally evocative and lovely sites. There's a lot to do in the Loire valley.

Their website mentions a new four star hotel at the abbey opening spring 2014. That may be the construction Tim saw in 2012.

If it gets too hard, how about Poitiers for the cathedral and palace? By train it's only about an hour and a half out and a bus runs from the station up to almost right between them.

we were just there last september. we stayed in a rented house just outside Fontevraud (maybe 5 miles) for a week. it is a VERY small town, with only a few restaurants open limited hours (however ranging from gourmet to a snack bar). there is however, an EXCELLENT patisserie there. didn't notice any construction going on. my suggestion/advice would be to stay in Saumur or Chinon and rent a car and make it a day trip from there, possibly seeing other things in the area of which there are many. it is a long way from Paris (several hours by train).