After Effects is a more extensive editing program, which includes several more options than Premiere - but Premiere does have quite a few transitions/effects to work with. I'm not sure if you are asking this question based on the fact that you don't have Premiere, or that you have it, but just haven't taken the time to explore the whole program. If you already have the program, and just aren't sure where to find the transitions/effects, you can see a list of them under Window ----> Show Transitions/Show Audio Effects/Show Video Effects. Hope this helped.
After efects is a compositing product, with no capture method. There is no way to work with After effects unless the clip is on you computer somewhere.