If I have an open source project that includes both BSD and GPL licenses as well as my own license, what will the overall license be or would different licenses apply to different parts of the code?
That depends of the combination of licences, up to the point that there may be no solution valid for all of the components and you have to rework your project. The terms from the different licences must be evaluated to see which terms are appliable to the final product.
For example, the GPL licence affects all of the source code of your project, and not only to the GPL licenced components1. So, imagine that you get one component under a licence which does not provide the source code (not even to you, even less for distribution) and a GPL licenced component. Incompatibility!
Keep in mind also that usually there are several "flavours" and versions of BSD and GPL licences (with some BSD licences being compatible with GPL while others at not), so there is no general rule.
That said, BSD licences are usually pretty open, so if you have a mix of (GPL-compatible) BSD licences and GPL licences, you usually can redistribute this with any licence that meets the GPL licence restriction. And, since the GPL forbids adding any additional restriction to GPL licenced code, that is pretty much it.
1That is why GPL is sometimes called a "viral" licence, because it "infects" all the code of your project.