It is tricky to get from Caen to Nantes by train. There are no direct trains. The best option (and there were only a couple per day on the random day (a Thursday) I checked on the SNCF website) is with a transfer in Le Mans. Both trains would be "ter" trains which accept bicycles. There is usually a hook to hang a bike by the wheel on these regional trains, and you do it yourself. There is no fee for the bike. On the day I checked, you could either depart Caen at 5:58am or 6:17pm. The latter would arrive in Nantes at 9:41pm with 17 minutes for the transfer in Le Mans.
The other routings all involve going to Paris on a train which accepts bicycles, but then taking a TGV train to Nantes. Only some of the TGVs takes bikes, you have to reserve it, and there is a fee. Plus it's a longer and more expensive journey.
Here's the website I used. I searched in French. I don't know if all of the same information will appear doing a search in English. If a train in the results accepts bicycles, there is a bicycle symbol which appears next to the train information. Generally, most (but not all) non-TGV trains accept bicycles and you load and unload the bicycle yourself. It's quite simple. The rail cars which takes bikes have a symbol on the outside of the car. On TGV trains, only some TGVs accept bikes, and you have to reserve it and pay a fee for the bicycle.http://www.voyages-sncf.com/billet-train