I am a self-published author of one book with another one in the works. If you are willing to do the leg work necessary to market your book (and usually any traditional publisher will leave it up the author to do their own marketing anyway) you will be able to make much more money self-publishing. The expense for the preparation of the book usually also falls to the writer, so you will likely be responsible for illustrations regardless of how you publish. I do indexing as well, and I know that many authors are now responsible for the creation of their own index. Publishing is changing!
Do an online search for agents. You will easily find many links that have lists of agents. Also do your research on publishers if you want to pursue traditional publishing. They will have various policies regarding the submission of manuscripts. Some will not take unsolicited ones, some will only accept book proposals, others will accept completed manuscripts, etc. If you can get access to a copy of Writers' Marketplace or Writers' Market it will give you some information and they also have information on literary agents.
I have not used the services of an illustrator but you may consider joining some self-publishing lists. They are good not only to learn about publishing, but also connecting with professionals of all types or to receive recommendations. You may also want to look at similar books to what you are wanting to publish and find illustrations that you like. Contact those illustrators if you can find contact info for them.
In Canada there is the Society of Children's Book Writers and Illustrators http://www.scbwicanada.org/
I would be very surprised if the US doesn't have a similar organization and of course you can work with people from anywhere in the world. These organizations might help to connect you to someone you can work with.
As for how you work with them, I think it is likely a very collabrative product. You certainly can approach them with your ideas, but you will also have chosen the person because you like their work and style and as a creative professional, they will likely produce better work if you give a basic concept but then let them do their job and insert their own ideas and style as well.
Good luck with your projects!!