Something I deal with all the time is trying to bring footage into After Effects from Final Cut Pro.
FCP to After Effects solutions.
There are several ways to go about this. If it's just one shot I'll often just bring the source file in directly and select my in and outs. Sometimes I'll render out a section of my timeline into something lossless, and sometimes if I need to do a lot of work and need the whole sequence I'll transfer the whole project with an XML file.
Here are a list of some great programs to make your life easier.
-Popcorn Island This is mostly what I use. It's a free script to import a basic FCP timeline into AE. This won't do anything fancy and it's a good idea to really flatten your timeline to one or two tracks first, but it does the job. And it's free. Get it here at Popcorn Island.
-Boris XML Transfer Version 2 If you need a better importer plugin for AE this is it. It really handles everything; audio levels, pans, gains, tansitions, FCP Native Filters, FCP Native Generators, Motion Tab Effects etc. If you need your your whole project moved over and need it to be true to your FCP sequence you might want to check this out. But it will cost you $299.00. Read more about it at here.
-Automatic Duck Another big one that's been around a long time is Automatic Duck. I personally have not used this one but I know several people who have. And they all love it. From what I've heard it does a pretty close to perfect job in bringing over your XML sequences to AE with little problems. Similar to Boris XML Transfer it also will bring over all the bells and whistles. They also have versions for Avid and Motion. It's a bit more pricey listed at $495.00. Learn more about it here.
So there you have it. Solutions to go from FCP to AE. Although Boris and Automatic Duck seem nice. I personally have never needed to bring in an entire sequence with everything perfect from FCP to AE. I usually just bring in a few clips here and there, or I'll bring in some simple sequences for color correction like you would with Color. Hope this was helpful.