Best thing to do .... start with the very basics and get your feet wet with some experience.
Seeing as how you would like to mix spoken word with background music and sound effects, and you obviously have access to a computer (lets hope you are on a computer that you at least own).

Start with what you have .... create with your computer.
First off you will at least need a half-way decent microphone (
Shure SM57 ), preamp (
Audio buddy or the likes ... even an old tape deck with mic inputs and record level adjustment can work) and a set of quality
isolation Headphones .
The next requirement, other than the soundcard for your computer, would be the software for mixing it all together.
For now go cheap (
n-Track ) or even free (
Kristal Audio Engine ).
Spend time learning the art of mixing the different audio tracks with whatever program you decide to use. Then do a Google search for royalty free music and sound effects or create your own.
Being as how this is a project involving kids, don't blow the mortgage payment just yet on gear. Main thing is to have fun with it and the quality time spent with the children.
A quality product will result from learning, understanding and ultimately experience.
Good Luck and enjoy.