I have many years of experience in Java including Swing, Servlet and JDBC, but have never programmed for a J2EE server.
Many job advertisements from large companies are specifically asking for J2EE experience. Are there specific skills or development environments that I should learn to qualify for these kinds of jobs?
Those jobs are most likely asking that you have experience with one or more of the major app server vendors, i.e. WebLogic, WebSphere, or JBoss. It's a bit different from what you're used to with servlets & JDBC, but it's still just Java. Typically you have to deal with things like JNDI lookups, dependency injection, messaging queues, maintaining application state between transactions, object/relational mapping, etc.
I have found that for most J2EE projects, it's more about knowing how components and systems are configured and how they work together than it is about any specific programming skill. And you also spend a lot more time waiting for the server to start up ;);
Download JBoss and get to work on the sample applications in the documentation. If you've done java, you're 95% there. J2EE adds the container and naming aspect to the java you already know and love. With the advent of EJB3, beans got a lot simpler as you only need a couple of annotations to get rolling with EJB. J2EE can be a bit daunting with the acronym soup of technologies available, but concentrate on the basics: EJB3, JNDI, JMS, data access (like Hibernate/JDO), and container basics.
get the book "Professional J2EE" from WROX press -- it is 6-7 years old BUT the content is pretty good. You should be able to get by just fine with that book and a 3.x version of JBoss or a 4.x version of JBoss and eclipse.
I think you're in good shape. Servlets are part of J2EE so you already have one skill. JDBC are used as well pretty much the same, but most of the time under some other frameworks.
So I would say you already have ( some ) J2EE experience. Read about EJBs, Transactions, and I think you're ready for an interview.
A lot of times the only way to get that experience is getting hands on. If you convince your interviewer that the skill you have already are enough as to get the position, you'll get the rest by experience.
Go on! You're ready. ( if not at least you'll learn from the interview :P )
Whenever you do start interviewing, Ken, be aware that the nutty Java marketing machine seems to be renaming J2EE to JEE. You could conceivably run across an HR department in the near future that has never heard of J2EE, but desperately needs JEE experience.
"Are there specific skills or development environments that I should learn to qualify for these kinds of jobs?"
If I were to interview someone for a typical J2EE shop, I would like to know how well you know the following
1) servlets
2) EJB (maybe)
3) JSP
4) ant
5) junit
6) subversion or other VCS
7) http and html
8) javascript
9) struts
10) hibernate
11) spring(maybe)
I'm not trying to scare you BUT 1/2 of what you need to know you can get by the "PROFESSIONAL J2EE" from WROX press. Rest of the skill, you should be able to get by with a spring book(most spring books also talk about stuts and hibernate) -- for example "The Spring Primer" -- http://www.sourcebeat.com/books/springlive.html.
Good Luck