Dan Poynter wrote a book on how to self publish, entitled The Self Publishing
Manual. It is available through amazon.com.  Poynter has for years been
involved in the small publishing community and I recommend his book although
I have not read that one.
Check Bil Henderson's The Pushcart Prize XXIV: The Best of the Small Presses,
Pushcart Press. Henderson also wrote a good history on the small press
movement in the US.

If you have had only three rejections so far, you have not even begun to
offer your work. Are you submitting to one publisher at a time or doing
multiple submissions (the way to go these days--just be certain you inform
publishers that you are doing a multiple submission).

In general, publishers do not like to take the illustrator along with the
writer. They exercise much more control by purchasing the manuscript and
assigning their own choice of illustrator, so that may be a small problem you
will have to overcome. Stick with it.

You can register copyright now. You can register as co-owners of the
copyright or you can copyright separately. You copyright the illustrations,
your author friend copyrights the manuscript. The publisher then has to deal
with each of you. You are wise to carefully lay out in writing what you
expect of each other, author and illustrator.

As a small press publisher I found the most work to be marketing. Especially
if it is a first book for both of you, you will have a choice of getting a
huge printing order (and a low cost per book) or a couple thousand or even
five hundred copy printing (and a higher cost per book). Then you try to get
one of the big distributors to handle the books for you.  While many self
published books end up being fuel for the fireplace and presents for
relatives, there are dozens of ways to market them. Think about selling some,
for example, to a bank or other  institution which will give them as
premiums. You get a wide audience you otherwise would not reach that way.

Be certain to get your ISBN number, and send review copies to everyone who
writes reviews. One good review can bring many orders, hundreds even and more
than you can likely fulfill on your own, and frequently a larger publisher
will seek to buy the rights from you if they see that it is a likely success.
That can be a lucrative way to go if you negotiate carefully.

Stay away from vanity "publishers" which are merely printers who do nothing
for you, despite what they say. And any book published with a vanity press is
a death sentence as far as being taken seriously by the publishing community
is concerned.

Self publishing has a long and respected tradition in this country. Many of
our known fiction writers started with self published books. Especially in
the late 60's and early 70's there was a very active underground press and so
many small publishers that there was an organization for them, called the
Committee of Magazine Editors and Publishers, complete with annual
conferences in universities.
Other resources: Cindy Shaw on this list can give you plenty of up-to-date
advice on small press publishing.

Hope these few notes give you some ideas. Good luck, JoanH.Lee