Have you looked into CreateSpace and Lightning Source? CreateSpace is easy on the budget, but I think LSI does a better job and lets you retain more control over pricing. Both are Print on Demand so you don't need to print lots of books up front. However, LSI does offer an option for short and larger runs if you DO want to print a bunch.
Both can do the full-color bleed that I suspect you'll want for a children's book.