I will try to categorize some different transitions, and comment on them. Of course, this is not carved in stone.
Yes, a cut is a transition. It sounds simple, but it has many flavors. The most simple is the direct cut, where you simply put one clip after another.
One of those flavors is the J-Cut, where you do not change the clip at the exact same time as the audio. For example a dialog between two persons, as a viewer you could turn your head to the other interlocutor after you hear he is now talking, so the audio changes before the visuals.
It is the cut that gives more continuity to a sequence, so much that actually you can use it to disrupt it, for example, you change from a battle scene to a calm paradise, this can surprise the audience trying to figure out what happened because they were teletransported without any warning. This is a smash cut.
But a cut can be back and forward between to shots, like in a dialog, this is a cross cut, or it can be to speed up a single shot, like a jump cut.
As you can see it has many variations.
The speed and rhythm could be different aspects of a cut. but you need to leave time for the images to be understood, the simpler the image the faster you can switch, the more elements it has, more time on screen it needs. Some music videos have so much frenetic rhythm that are unwatchable.
This normally implies some calm situation, it can add an "elegant" element.
They can be used between clips or between two bigger sequences. It can either link them or distance them depending on the context, but normally it links them despite an intrinsic element of distance, for example, one person thinking about some other across the world.
But it also can show the passing of time.
The duration of the transition itself is another narrative element.
Fade to - fade from
Black or white.
A fade to black is one of the transitions that distance two senes the most, it gives a sensation of closure, for example when some years have passed and the prior situation wants to be forgotten.
A fade to white can have a similar effect, but gives a small relief, implies some hopeful feeling. It can imply some "metaphysical" element, passing by, new beginnings.
It can be used between two scenes or as a final element of a video.
It also has some variations depending on the "blending mode" applied. If a transition has a blending mode that increases the luminosity of the video can be used as an intro to a flashback or something like that. But this variation could look "cheap" as a low budget tv show.
This three are the most often used by far.
One element appearing above the two clips as a middle transition element. One example is a "film burn", where this foreign element appears and disappears during the transition. Normally it implies a different blend mode different than just a dissolve.
A film burn is can be used of course to give some nostalgic feeling, but you can be creative.
Normally this combined with a camera movement, for example, a camera travels from left to right and a tree is very close to the camera, and now another scene is revealed behind a light pole with the same camera movement. (It is a creative way to use a wipe)
You can approach to a black object, and start the next scene walking away from another black object.
It makes an element appear from one side overriding the image underneath. If it is used between clips, it can give a "comics" style. It can imply that the two scenes are occurring in different locations at the exact same time, for example, a telephone call.
But it can also be used to add an element over the existing scene, for example, some text over a video, or an enhanced version of a video over the original.
The edge could be sharp or can have some gradual blending.
Changing focus, or adding some kind of a gaussian blur.
An obvious usage is a person losing conscience but can be a cheap... or economic way to switch between scenes. It was used when direct cut was the only available tool, turning off a recording and on again. I doubt is used a lot these days.
They need to be combined with other resources
For example with camera movements, lighting conditions, and above all... the story.
But you also can use whatever you have!
Yes, you can throw a "fancy" transition on a kids video, with balloons, stars, and flipping pages. Just keep in mind that this can have a very short-lived style. But probably this is what you need, giving a video a 90's look could be what you need.
Keep in mind that "Video editing" and "Video compositing" are two different but related crafts. Some motion graphics or visual effects elements can be considered transitions.
A video being torn by pieces, color bars floating around, geometric shapes, particles appearing, maskings, overlays, 3D movements, glitches... Just remember to have the elements with a context in mind. It is different to make a music video for a Techno music group than a documentary.