We will be going to Versailles in June and spending the day there.
Is it better to buy tickets online or when we get there? We won't have a museum pass.
Is it worth the extra money to see the Royal Serenade and the Musical Fountain Shows?
We have no time constraints.
I can't speak to your second question as I'm unfamiliar with them, but as for the first, when we went to Versailles this past January with visitors from the U.S. we found that having pre-purchased tickets saved us easily 30 minutes or more with the entrance time, as we were able to walk past the (long) ticket line and go straight to the (relatively short) security line.
I can't say how an experience in January will translate to your time of visit, but if it's during the main tourist season pre-purchasing tickets probably would help even more.