Specifically, primarily for children’s picture books (mostly illustration driven, with short descriptions of the subject matter.) I have no idea what the going rate is for writing; I usually only do illustration.
If you’re already an illustrator you should look into The Society of Children’s Book Writers and Illustrators. They have publisher information, conferences, publications, chapters, the whole works.
And the answer to your question is the age old: it depends. There are big publishers and small ones, and dozens of different varieties of books calling for various types and sorts and kinds of illustrations.
I was just jumping in to make the same recommendation as Exapno. It’s worth joining SCBWI just to get the membership packet with its information about standard contracts, royalty rates, and so forth.
The two standard models for illustrated children’s books are for the writer and illustrator to split the advance and royalties, or for the illustrator to get paid cash (no royalties) and the writer to get the advance and royalties. Advances range all over the place, from nothing on up. Somewhere between $2K and $8K seems common. Royalties can be anything from 5% to 10%.