What's possible is just about any effect or graphic you have ever seen or dreamed about. After Effects is incredibly powerful.
If you are new go to the Adobe After Effects product page and click on the links in the Learn section. This will give you enough of the basics to get started.
If you create your illustrations in Adobe Illustrator you should also use the search help field at the top right corner of AE to search for things like "preparing illustrator files for After Effects" and "Import AI files as a composition."
If you create your artwork in Photoshop it would be a good idea to search for similar topics but use PSD or Photoshop instead of Illustrator.
To do much more than just add a glow or some distortion to your artwork you need to separate it into layers.
There is no easy path to complex and interesting effects. You have to just start going through the basics so you learn about things as simple as frame size and frame rate. Note: Until you become an expert your composition setting should never say Custom. Use the presets and I would strongly recommend that you use the HDTV presets and and avoid like the plague any preset that does not use square pixels.
Then it is just a matter of imagining what you want to do with the lines and colors and experimenting with the effects that come with AE. They are categorized so it's pretty easy to start going through the categories and figuring out what each of them do.
I hope this helps. You could easily spend an entire masters program on AE and not learn all that it is capable of so be patient and start small. Maybre figure out what you can do to a single curved line first and then go from there.
I think it's very complex problem for newbie. If you want to do it on your own you should better know basics of AE as minimum. There is actualy no button "Make it perfect". Even with third plugins. See Plexus, Trapcode Particular, Trpacode Form for example. But with plugins you should know them in advantage level.
Thanks for your detailed reply and tips Rick.
Thanks Disore. I was never seeking "perfection"