I am trying to familiarize myself with JavaEE. I am a bit confused as to what the purpose of each "component" (for lack of a better word) is: Session Beans and Servlets, and how they properly interact with a web application (client-side JavaScript).
In an attempt to understand this I am building a simple web application. What is the preferred way to use each component to build something similar to the following:
When sending the request, do I send it to a Servlet (which uses an EJB), or to a Session Bean through WSDL? How do I go about maintaining a "state" for that user using either method? I assume with Session Beans it's as simple as annotating it with @Stateful.
Also, I assume the requests sent from the client side must be in SOAP format. How easy is it to use something more lightweight (such as JSON)? While I would prefer to use something lightweight, it's not necessary if SOAP makes development faster/easier.
The Java Enterprise Edition tutorial address pretty much all of the topics you bring up; what's the purpose with the different kind of bean types, how do I implement web services, how do I implement authentication, etc.
I highly recommend you take the time to build the sample application, especially if you're completely new to the Java Enterprise Edition (Java EE). It is important you build up a good understanding of the core concepts because it can be hard to know what to focus on in the beginning due to the breadth and depth of technologies and standards that comprise Java EE.
One thing to keep in mind is that while Java EE certainly tries to support best practice and enable design and development of secure enterprise applications that perform and scale well, it does not prescribe or limit enterprise applications to follow one particular protocol, data format, and enterprise application design pattern. Some protocols and formats are better supported out of the box by the core framework implementations, and some choices are vendor-dependent, but very few specific technology choices are locked into the specification.
To answer some of your specific questions, Java EE has great support for SOAP, but it does not preference nor limit web services to the SOAP protocol. With JAXB and JAX-RS it is just as easy to develop RESTful web services that accept and return XML or JSON, or both. It's up to you to decide whether you need to use SOAP, REST, or another protocol.
It's also your choice whether you want to use frameworks like JAX-RS or explicitly develop Servlets to handle HTTP requests and responses. In many cases, JAX-RS will have everything you need, meaning you'll be able to implement your web services as plain old Java methods with a few annotations without ever having to bother with marshalling and unmarshalling contents and parameters.
Similarly, with JAXB it's up to you whether you want to use WSDL or not. It's great if you have WSDL definitions, but no problem if you don't.
In many cases you will typically maintain state using the Java Persistence Architecture framework (JPA), and access and manipulate such data through stateless session beans. Developers new to Java EE are often tempted to use stateful session beans to maintain state that is better managed in the persistent storage. The tutorial takes you through the different kinds of bean types and their purpose.