So yeah, just keep it simple and everything will go just fine.
LibGDX is not simple. LibGDX is a complex library.
LWJGL is simple. It is just a wrapper for OpenGL.
There is a difference between simplicity and ease-of-use.@OP:
No-one here is going to recommend using Swing/Java2D for games. It's just not designed for it. As far as learning how to get things moving on a screen goes, it's great, but making a games with it is not worth the struggle. So instead, everyone will suggest to you that you use LibGDX.
LibGDX is fine. It's a library which eliminates having to write boilerplate OpenGL code, and it also includes lots of helpful libraries for gamedev. It also gives you the ability to (relatively) easily export to HTML5 and Android, if that's what you're looking for. Unfortunately, if you plan at any point in the future to move away from Java, or if LibGDX goes under like Slick2D did (it's not very likely, but it is possible), then all your knowledge of LibGDX will be useless. Also there is the off-chance that you end up wanting to work on a team project that doesn't use LibGDX.
So, I will tell you that there is the option of LWJGL. It is a library that wraps OpenGL functions so you can use them in Java. LibGDX in fact, is based on top of LWJGL (or JOGL, if you'd prefer that). Let me warn you that OpenGL is a very simple
library, and so you will need to write lots of functionality yourself if you choose to go down that path. However, it is a very common standard and most general purpose programming languages have bindings for it. Also, many libraries are based on LWJGL (as mentioned before, including LibGDX), and so you can pick those up quite quickly as well if you can see what OpenGL calls are being made under the hood.
Then there is JOGL. It's similar to LWJGL, in that it's an OpenGL wrapper. The library design is somewhat different though. The majority of active people on this forum aren't familiar with it, but there is another forum somewhere... Anyway, I'm mentioning this because if I don't, a certain member on this forum may take offense.
There is also Slick2D, but we don't recommend that anymore. In fact, the developer behind it (kevglass) has switched to LibGDX. Apparently it's still being maintained, but it's been a while since I've heard anything about it.
So there, you now know all the (common) options. You can choose based on what suits you.