"How do you find information on unmarried mothers and the effects it has on the children"
With the types of issues you're describing, a good search to start with is "Single Parents" as a subject in the PsychInfo database (click to view search). A lot of those articles have to do with children of single parents and issues they might develop, possibly as a result of coming from single-parent households (though you have to be careful with claiming causation, e.g. "effects", as a finding of these studies).
Also check out this CQ Researcher piece on child poverty, and whether unwed mothers are the cause.
Please let us know if you have further questions.