It's a bit of a Catch 22 situation. Most publishers will not accept "unsolicited manuscripts" and only accept manuscripts recommended by literary agents. Most agents agents don't seem to accept authors who haven't been published before. See the catch?
Now fortunately I have been published before, but if you haven't getting your first publication must be really daunting. I'm lucky that my editor from my previous publisher (who is not the right publisher for this project) is going to point me in the direction of some contacts she has. I'm also doing some footwork myself.
Interestingly all the publishers and agents' websites say (often in bold) "we do not accept unsolicited children's fiction". I guess that's the Harry Potter effect, they must be buried under mountains of paper for the next wizard franchise.