Hello. I'm creating a match3 like game and I wanna that my game has visual effects like on images below. 

My question is how can I create such type of effects? Is this a frame-by-frame animation or well written shader? Or this can be created using some particle system?

I'm a newbie in this field and I really do not know what I should start learning to create such type effects.

Here is a video of a cool Match3 game. If you set playback speed on 0.25 you can see how rockets and bombs explodes. May be this help you to understand

https://www.youtube.com/watch?v=3gjAOF-rc1s

Below I attached screenshots of a bomb explosion

52 minutes ago, Alexander Nazarov said:

My question is how can I create such type of effects? Is this a frame-by-frame animation or well written shader? Or this can be created using some particle system?

I doubt that it is a (pure) frame by frame animation. If you have a look at the borders of the pressure wave, you will see that the neighbouring items are visible/mirrored through the wave.  So this seems to be a post-processing (shader) effect. The fireball might be a frame-by-frame animation but a lot of other techniques are also thinkable. The shards are probably just managed by a particle engine. As you can see on the last image, the violet shard in the centre beneath the blue box looks quite similar to the shard located in the lower right corner (the one that lies on the border of the pink surface). If you look closer, you will also find rotated versions of this shape elsewhere.

1 hour ago, Alexander Nazarov said:

I'm a newbie in this field and I really do not know what I should start learning to create such type effects. 

Most basic effects are quite easy to understand and start looking super impressive if you combine them. So maybe you start by just creating a small particle system that throws some debris into the environment.   You can then use a modified version of this particle system to create the fireball. Afterwards, add some light effects. Finally, try to build the shock wave. I think this is the most complex part, but I might be wrong. ?Greetings

If I understood @DerTroll right, they're saying you can use multiple techniques on the same effect sequence, which I agree.

You can combine particle systems, frame-by-frame sprites, shaders etc. to create one rich layered effect.

An element of the effect like that shockwave could be done with a particle system with a single particle, and only scale animation and refraction shader. This way you could use the sane particle system code to animate both sprite cloud elements (sparks etc.) as well as large single elements (one shockwave), by using different p.systems for each.

For inspiration, some frame by frame FX by AlexRedfish (https://www.deviantart.com/alexredfish/gallery/46778529/FX-Animations) and an example of a 2D particle editor for games, TimelineFX (https://www.rigzsoft.co.uk)