If you are interested by Java EE, I would warmly suggest to start directly with Java EE 6 which has been officially released last week. Just go to http://java.sun.com/javaee/ and download GlassFish v3. For the integration with the Eclipse Java EE IDE, simply install the GlassFish Server Adapter. You'll find plenty of samples for Java EE 6 at http://java.sun.com/javaee/reference/code/ that you can also get through the GlassFish Update Center (just do it). They will definitely help you to get started.
In your case, my advice would be to start with Java EE 6 "only" for now. Get familiar with EJB 3.1, Servlet 3.0, JPA 2.0 (the Java Persistence API, Hibernate being one implementation), JAX-WS, JAX-RS, maybe JSF (this may still be controversial but I think that the new Servlet 3.0 API makes any advice about web frameworks irrelevant so... I won't make any recommendation here). Leave Spring alone, you won't need it in early steps (you may not need it at all) and there is enough to learn in Java EE 6 itself. If you want to learn Spring, for example because it is widely used in Java EE, at least prior to Java EE 6, you'll still be able to learn it later. But don't worry, Java EE 6 is very impressive and you can do really nice things with it, without the need for Spring.