AI am a beginner in the cake world, and I am making my son a Mickey Mouse cake for his first birthday. I was just wondering how much icing is okay to put underneath fondant to make it taste better for the kids.

You will get different opinions on this.  Some just crumb coat then cover.   Because most people in my area do not like fondant and peel it off...I completely ice my cake before covereing in fondant.  That way when they do peel off, they will have a buttercream piece.

ASo if use more icing should I put the cake in the fridge to firm up the icing before I put the fondant on?

As much as a cake that doesn't have fondant. 

I agree AZ....I know lots of people that only crumb coat.   I've seen it done that way on tv also.  

OP.. I completely chill my cake (overnight) after I ice before I fondant it.