A video is just a sequence of images invoking the illusion of continuous motion, so there are many ways in which an artist can use video in their work, whether or not they would consider that to be Video Art.
I will discuss making video with a camera to begin with but will then discuss making art directly on the computer without using raw footage.
The simplest way to make an art video is probably to make a slideshow of still images. Whatever system you are using (PC, tablet or phone) you will have an app that lets you stitch photos together to make a slideshow.
It is simple enough to use the same process to construct a slideshow of your paintings. On a mac or an iPhone you will have iMovie (and something similar on Android) and on a PC you will have (or can download for free) Windows Movie Maker. Both of these apps allow simple editing. You can convert still images to short movie clips, You can trim clips and join them together in a sequence. You can add titles and audio. You can apply selected filters, such as making an image monochrome. Filters differ from app to app.
"It's not that simple ..." is actually an attempt at Video Art. The images themselves are not the main object, it's the sequence and the synchronisation and the choreography that are the main object. This was originally constructed on my phone using iMovie and refined on my PC using a more poweful editor. The phone version was fine, as a sketch. the more powerful editor was needed only to get the timing exactly right.
The images in that video are photographs that have been post-processed to be strictly black and white (i.e literally just two tones).
Moving to a more elaborate video editor is only required if you need to be more ambitious in your effects. I use Final Cut on a Mac. The equivalent on a PC would be After Effects and again tablets and phones have a wide variety of apps to go beyond just the basics.
The video effects that I find most powerful are all in the category of "blending effects", where two or more videos are merged in some way to make a collage. For example, blending a video with a delayed version of itself will give you two images on the same screen where the motion of an object is shadowed by a copy of itself a couple of seconds behind. For example, a dancer can be made to dance with themselves in this way.
Blending many videos is straightforward so that many images are collaged together. One very useful way of using blending is to create a stencil (also called a mask or a matte) through which one video is printed onto another. If the stencil is itself a video, then the effects that can be achieved with experimentation are beyond the simple collage created with blending.
A video editor is, like a paintbrush, just a tool for making art. In fact, you can make Video Art using only an editor, which is what I will discuss next.
A video editor will normally allow you to do things like create simple geometric shapes and combine them into an image. The shapes can be made to follow paths on the screen and to pass across each other. So you can see how it would be easy to make a composition that would have the appearance of a geometric abstraction with motion.
The best way to achieve this is not to combine the shapes on one clip. Rather, create separate clips for each shape. If the shape on that clip is made to follow a closed path then a video loop can be created, so that a longer clip can be made by assembling many copies of that basic clip.
The idea now would be to make many clips with different shapes following different paths and to blend them into a composite. By devices such as making the shapes transparent, by varying the duration of the clips and by varying their speed, complex choreography can be achieved.
Of course, few artists actually would be satisfied with making geometric art, but it is an excellent place to explore the potential of video editing as a creative tool. Combining geometric animations with shot footage is a direction that would allow any artist to move towards their own style in video. Also, video editors will always provide a wide range of video-effects, such as picture-in-picture, tiling, fish-eye, colour manipulation, speeding-up, slowing-down, reversing etc as well as adding and manipulating audio that gives video a dimension beyond what is achievable with a simple slide show.