I would note that I have found no drop-in replacement for Premiere.
Cinelerra is the one people usually recommend, but on my system it is unstable to the point of being useless. There is a program called kdenlive which looks like it might have promise but still feels kind of underfeatured.
There is Kino, which is a strange editor, but is the one I've had the most success with. Kino is fairly stable and dumps DV pretty well if you have a DV video camera, but its interface won't be what you're used to.
Open Movie Editor looks promising, but I've been do disgusted and put off by the state of video editing in Linux, I haven't had the energy to try it. If you do, let me know what you think.
You will almost definitely want to have avidemux, which is a sort of swiss army knife for video format conversion and the like.
This is a personal opinion, but I'm not sure I'd look for a dedicated video editing distribution. Rather, I'd find a solid and well-supported one that has these packages in its package manager. A lot of the niche distros tend to be under-supported, or short-lived (I suppose there are probably exceptions somewhere), but I'd rather have a Debian or Ubuntu as my OS, with video editing software running on top of it.