I've used Premiere since its non-Pro days, switched to Final Cut Pro for a few years, and am now back fully into Premiere Pro CC. The biggest advantage for me from an aerial footage perspective is the ability to import footage straight from the GoPro without transcoding first. There's a minute or two while it indexes the footage, but after that, you can dig straight into previewing real-time and cutting.
Premiere is definitely a higher-end editing tool (as compared to iMovie or Windows Movie Maker), and has a slightly steeper learning curve, but it's a hugely capable piece of software. I make pretty extensive use of After Effects CC for more elaborate manipulations and compositing, and even Photoshop can open and process/manipulate video footage. The point is that if you get the full Creative Cloud Suite, you have an enormously powerful post-production tool in your hands. I haven't used the most recent version of FCP, and it may very well have a lot of the same features, although the other tools in the CC Suite make it a better deal for me.
The Warp Stabilizer VFX plug-in does a pretty amazing job not just stabilizing jerky footage overall, but will smooth out any pans, tilts or other camera movement that you attempted in the shot but didn't get.
I'll agree with Erotic Panda*, there's no real substitute for correctly shot footage. The less you have to do in post production, the better your final product will be. Ideally you'd want the 3-axis gimbal flying in ATTI mode on a GoPro with ProTune turned on for the best results on the back end. In a lot of cases you will need to do more work in post-production to get a good-looking image, though. ProTune looks a lot flatter in color right after you acquire it, but it gives you a lot more leeway when doing color corrections in post, since it's not compressing as much or doing ham-fisted, one-size-fits-all color modifications while recording.
Having said that though, you can get away with quite a lot in post-production these days before it becomes distractingly noticeable. I prefer to shoot in 2K on the GoPro 3+ Black with the goal of outputting to 1080p. This gives a lot more flexibility when doing any optics compensation or stabilizing, and offers the ability to crop the image or slightly change the framing mid-shot.
Stabilization will always result in less useable image area from your original frame size. On a raw stabilized shot, you'll see the frame edge creeping into the shot, and most of the stabilizers will zoom in to compensate and cover this up. This means that you'll lose sharpness, because it's increasing the pixel size. If you shot at 2K (or 4K on the newer GoPro 4 Black), scaling down to 1080p/2K will reverse any sharpness issues (to a point) introduced by the stabilizer and/or optics compensation. Ideally you'd want to tell the stabilizer to not do any zoom/scaling compensations, and avoid the upscale/downscale roundtrip. Just crop the image down to your final resolution after it's been stabilized, but this is something you can play with to see what works for you.
Still, like I said, the better footage you start with, the better your output will be, and there will be some instances where no amount of post-production tweaking will fix a shot, but there's a huge grey area in between that will give you acceptable, if not great, results.
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* Add this to the list of phrases I never expected to say/type.