With LWJGL you have access to OpenGL and control over all the fancy hardwae accellerated stuff. Slick utilizes this while providing an API vaguely similar to Java2D, so is a good choice for stepping up from Java2D to LWJGL (you can directly use your own OpenGL code in conjunction with Slicks provided functionality)
Regarding the applet installation, there are ways to make this as smooth as possible for the user, but there is always the downside of a security warning and a higher possibility of something getting wrong due to the complexity of the matter. But it seems to work quite good - at least for all the minecrafters out there. It will however require you to dig into this (no pun intended
For Android development, take a look into LibGDX. You can use OpenGL ES with it and it has a LWJGL backend to test (and deliver) on the Desktop. There is even an (experimental) LibGDX based Android backend for Slick2D.
So even if you don't switch over, take a look at Slick2D, LWJGL and LibGDX.