After Effects can be used for TONS of things involving compositing and motion graphics. It's quite powerful. But it can be a bit overwhelming to a novice. If you're considering getting into AE, I would suggest the excellent
Video Copilot tutorials. They cover a lot of the basics and they're free.
Premiere Pro would be best compared to Final Cut Pro and Avid MC, being a non-linear editing application.
As mentioned before, there are a number of included (and available 3rd-party) effects available in the aforementioned NLE packages, but their functionality and flexibility are limited in comparison to what you can do in dedicated motion graphics applications.
Apple's competitor to After Effects is Motion (included with Final Cut Studio). Motion is a fairly powerful application in its own right, but AE really is the industry standard and has a much larger support base. I do my actual editing in FCP (personal preference), but post effects in AE.
AE does have limited timeline editing capabilities, although I'd strongly urge against using them for a couple reasons. For one, it isn't exactly intuitive for that purpose. Secondly, it can cause workflow issues with related applications. So, do the actual editing on the NLE of your choice and the post effects in a motion graphics application.
