You might already know this but they are one in the same except that Java EE is the enterprise version, meaning more advanced features such as servlets.
I would say learn everything in SE (core) java before going into the Enterprise Edition. Now it's not enough to learn something to know it, you should understand it as well.
For example: Why does Java have Multiple Inheritance in terms of implementing multiple interfaces rather than extending multiple abstract classes like C++?
Why would you use and Abstract class over an interface and vice versa?
I'd say gather as much of an in-depth understanding to the core aspects of java before tackling the advanced topics. Also conquer data-structures, Object Orient Principles, Generics, Threading, annotations, and anything else that may envelop the basic understanding of java. Now of course you don't have to know all the intricacies of java just have somewhat of a basic understanding.
Scour forums and EXPERIMENT! Code non-stop until your fingers bleed. Lastly always ask questions to what you have learned!!!!
Any fool can know. The point is to understand.