I am new to fondant cakes and my fondant begins to melt when I take my cake out of the fridge but I'm afraid that if I leave it out, my buttercream will melt.  The cake on my profile picture began melting and the Minnie ears broke off.  My daughter was not happy.  Can someone help with the proper steps to take?

What do you mean when you say it melts? If it's humid when you take the cake out of the fridge, moisture from the air causes condensation on the surface of the cake, but over time, it will dry out again if left alone.

If where you live is very humid, there's not much to be done about that, except you can box up the cake and wrap in plastic wrap before fridging it, and not unwrap until the condensation dries. Others have better advice about that process, I haven't done it myself.

Otherwise, you can try a different kind of buttercream or ganache that can be left out of the fridge so you don't have to put the cake in the fridge at all. 

What are you attaching things like your minnie ears with? If it's just with water, then the condensation might dissolve that connection. You might need to try using royal icing or a gum glue.

I live in Florida so it is very humid.  I'll try boxing and wrapping it next time. Thanks.

To make the ears I used fondant with some lolipop sticks inside for support with some of the sticks sticking out so can attach to the cake. But the fondant broke alongside the sticks when they got wet from the condensation.

Did you add any tylose to your fondant? Fondant alone isn't really strong enough to hold up in those conditions. Or you could use gumpaste.

This is BlakesCakes's method for freezing cake - I'm assuming it would work to fridge them as well.

Box completed cake. Wrap box in at least 2 layers of saran, one layer of foil. Freeze. 24hrs. before display, place in fridge still boxed & wrapped. At least 2 hrs. before serving, place still boxed & wrapped cake on counter to come to room temp. When ready to display, remove wrapping.
 

@ mcaulir -  thank you for referencing this method by BlakesCakes.  I also live in a humid climate and about to cover my first ganached cake in fondant.  I have read so many threads in debate on whether to refrigerate a covered cake.  I would really prefer to do so in my case and thought the cardboard box was my only safe guard.  Never thought to add extra layers of protection by wrapping that box up in plastic wrap.  I won't be freezing my decorated cake but I think this technique is probably my best bet for bringing the cold finished cake to temp in an air conditioned room.  It definitely sounds like gradually bringing the temperature up is the key to success.  And hopefully if some condensation occurs it will dry as most note.