Video or movie production on Linux seems almost limitless thanks to Hollywood's adoption of this great, stable, platform. I would say professional compositing, animation, and special effects is more present and mature on Linux than home friendly video editing software. This is changing however, perhaps as soon as later this year, we could see the release of OpenShot 2.0 which I anticipate will be the most user friendly and intuitive video editing software suitable for the beginner, to use at home.
I did my first video editing in high school on Windows XP with Windows Movie Maker. I liked it, it was intuitive and easy. I found OpenShot 1.x while I was looking for something like Movie Maker for Linux and I have to say it is my favorite video NLE (Non Linear Editor). It is easy to layer video or images on top of other video or images. It has some transitions for transitioning between clips, some useful effects to apply to the video clips, and is easy to do basic arranging of clips with drag-and-drop. OpenShot 1.x is good but just isn't there because of bugs, stability, and a few missing features. OpenShot 2.0 and beyond should fix these shortcomings for the beginning home user.
There is nothing, however, keeping a determined amature film creator from learning and using some of the most advanced Hollywood video software on Linux. Notice I say 'some' because much of what Hollywood uses is custom 'in house' software, but there are complete commercial and open-source packages that are free to use for the sake of education or even professionally.
Lets go over what the film creator wanting to go pro can dig into. A lot of professional movies have SFX (special effects). All SFX are done through video compositing. Video compositing is the process of combining separate visuals into a shot. This often gives the perception that everything seen is part of the same scene. Perhaps the most powerful program to do this is a compositing package called Nuke. Nuke was used in many films including The Hobbit, Avatar, and Tron: Legacy. Nuke is a node compositor which is considered more suitable for complex compositions than the other type of compositor, called a layer compositor. A popular compositor by Adobe named After Effects is an example of a layer compositor. There is a free version of Nuke available for Linux which is intended for non-commercial use, you can download it from their website here.
There are other options than Nuke for doing your video compositing. An open source alternative to Nuke is Natron. Natron is compatible with the industry standard OpenFX plug-in format which is also used by Nuke and other commercial compositing packages. Many packages that do 3D computer animation can also do compositing. Houdini is a popular Hollywood example of one such software that lets you do both. Houdini has a free educational version called Houdini Apprentice the next cheapest version I can find on their site is Houdini Indie version for $199. Another extremely popular animation and compositing software is Autodesk Maya. Maya runs under Linux but I have found no free version for Linux.
If you are looking for an open source alternative to Houdini or Maya, look no further than Blender. Not only does Blender do 3D modeling, animation, compositing, and simulation but it also has a non linear video editor built in. The video editor is very capable but there is a catch, you must use video clips that all have the same frame rate. Blender has only recently been started to be used in Hollywood for small enhancements to scenes. Blender has also been used for some movies on Lifetime Movie Network and on History Channel. Blender allows for any property to be animated including those found in the video editor. It also can do 3D video tracking to insert 3D objects into the camera scene. Blender has a node compositor with many nodes built in. There are plans to support OpenFX in the Blender compositor but the project is not currently finished. If you like intro titles for your video you can find a lot here.
What I have talked about so far should give you a good idea of what to focus on learning if you want to create SFX for your video. Now if you want to arrange clips of your finished scenes you probably want a good reliable video editor. The most reliable stand alone video editors on Linux are probably Lightworks and Cinelerra, both are used in Hollywood. Cinelerra will run slow if it isn't setup properly. I have only had Cinelerra run smoothly for me one time by accident. Lightworks as of now is probably your best option for a professional video editor. It is designed to easily manage a ton of footage if you are thinking of making a full length movie with lots of camera angles and such. There is a free version which is planned to be open sourced in the future. You can buy or rent the Pro version for $24.99 a month or a one time payment of $437.99. The free version limits export to 720p. Novacut is another promising video editor for full length films.
I may write more in depth about these softwares once I get projects where I can use them more, but as of now you will just have to try them out to see what will suite you and your project. Some video editors worth keeping an eye on include: slowmoVideo, Pitivi, Kdenlive, LIVES, Shotcut, and Flowblade.
EDIT: Piranha 8 looks like a very professional video editor if you can shell out the $1000 bucks for it. The list of films it was used with looks impressive.
