Hi Blue Moon,
As Heather mentioned, the Children's Writers and Illustrators Market is a great resource for finding potential agents, publishers, etc. The SCBWI publication is available online to SCBWI members. Joining SCBWI is great -- it's a huge wealth of information.
If you plan to submit to publishers, no, I would not self publish it in the meantime. Most publishers do not look at a book that has already been published. (Sure there are a few notable exceptions but in general, one chooses whether to go through traditional publishing or self publishing.)
Submitting to a contest may be okay, just be mindful of the rules of the contest as well as the publisher's own guidelines. For instance, if the contest intends to publish your work or any part of it, just be aware of that. If you ever have questions about a specific contest, post the contest guidelines here on the SCWBI blueboards and I am sure you will get some wise feedback about whether it should be okay to submit to that contest while subbing to publishers, or not.
You might also want to have your work critiqued, if you have not already. You can find peer critique groups by joining SCBWI.