I'm totally new to AE. What are some things typically done to clean up video and make it look nice?
Common postproduction work:


Color Correction
Reframing (crop out unwanted stuff from images, especially viable if shot in a higher resolution then distribution format)
Stabilization
Speed Changes and Speed Ramps


Personally i doubt that you need to use After Effects for this though, as all of that can also be done in Premiere. If you want to learn After Effects in the process, i'd look into making some nice typography animations which you later import into the final video. There are tons of tutorials on that on the interwebs and / or you can download some After Effects templates to get you started.