I have quite a few ideas in mind of options to take after I am done primarily using swing and awt. But, how long should I stick with it? I feel ready to move onto LWJGL or JOGL.

If you are serious about game development then AWT/Swing will only take you so far before you start hitting its limitations for anything slightly demanding.

Switching over to OpenGL is a good idea as you can pretty much throw anything at it and it'll render it decently. Do keep in mind both LWJGL and JOGL are just bindings to the raw C libraries which are pretty low level, so you'll spend a lot of time writing boilerplate code and reinventing the wheel, so going for something slightly higher level will help you be a lot more productive.

Libraries like Slick2D or LibGDX are a good first step up from Java2D as they give you a nice easy high level API (similar to Java2D) while giving you the speed and smoothness of OpenGL (and OpenAL).

Some engines support both LWJGL and JOGL, for example JMonkeyEngine 2.0 (contact me in PM if you need some support of JMonkeyEngine 3 for JOGL) and Ardor3D. That's true for NiftyGUI too (I wrote its JOGL 2.0 backend). TWL only works with LWJGL but it would not be that hard to port to JOGL thanks to NEWT. JMonkeyEngine has a large community and a nice game development environment based on Netbeans RCP but I find this engine less reliable than Ardor3D.

JOGL has a few build-in widgets but they require shader support  

I agree with kappa, LibGDX and Slick are worth a try too.

Agile2D is very similar to Java2D.

As far as I know, you can use Swing components with JOGL and LWJGL.

Maybe look at this too:
http://www.java-gaming.org/topics/hello/24411/msg/205141/view.html#msg205141