This is a really broad question, but there are two tools which can help you to create unique graphic effects in Unity:

* [Programming your own shaders][1] gives you a lot of freedom to develop graphic effects on the GPU
* The [Mesh class][2] gives you access to the vertex and UV coordinates of your 3d models. You can use it to deform or even generate new 3d objects at runtime.

And then there is of course a lot of cool stuff you can do with the special effect tools Unity provides to you out-of-the-box:

* The [post processing system][3] can add all kinds of neat visual effects to the whole camera. You can even [program your own][4].
* [Particle effects][5]
* [Line Renderers][6] 
* [Trail Renderers][7]

If I would want to copy Duet in particular, I would do the trails of those red and blue balls with trail renderers. For the after-image of the blocks I would try motion blur first and when that doesn't give me the effect I want I would use particle effects to generate fading clones of the same sprite used for the block.


  [1]: https://unity3d.com/learn/tutorials/topics/graphics/gentle-introduction-shaders
  [2]: https://docs.unity3d.com/ScriptReference/Mesh.html
  [3]: https://docs.unity3d.com/Manual/PostProcessingOverview.html
  [4]: https://docs.unity3d.com/Manual/PostProcessingWritingEffects.html
  [5]: https://docs.unity3d.com/Manual/ParticleSystems.html
  [6]: https://docs.unity3d.com/Manual/class-LineRenderer.html
  [7]: https://docs.unity3d.com/Manual/class-TrailRenderer.html