Are Swing applications really used nowadays? I don\'t find a place where they are used. Is it okay to skip the AWT and Swing package (I learned a bit of the basics though)?
You may checkout Swing Sightings.
This website is hosted by SUN and it is dedicated to sw projects that use Swing. There are a lot of projects using Swing ...
http://java.sun.com/products/jfc/tsc/sightings/
If you are writing for the web exclusively, you can probably skip Swing, but otherwise you're absolutely going to run into it. I've never worked on a non-trivial Java app without a Swing GUI.
Also, Swing is one of the better APIs to use. If you use most others, you are going to find them more difficult to use and/or platform incompatible. (If anyone reading this is aware of exceptions to this, please leave comments, I haven't looked for a while and am kind of curious if anything better has become available)
Other JVM languages like JRuby and Jython are often used because of their access to Swing.
You should probably be looking at JavaFX for future work. As the FAQ states,
Sun is not replacing Swing with JavaFX Script. Instead, JavaFX Script makes Swing easier to use.
Using/learning it will give you a very good understanding of the important Swing concepts, and how they fit in with "the future". I haven't used JavaFX yet, but heard very good things about it.
Whenever you want to write a desktop GUI app for non-Windows OS, you use Swing as the only sane and guaranteed cross-platform GUI framework.
Besides, Swing has a very good design. I recommend to learn it anyways, to make yourself a better programmer.