I would like to test this app, to see how it actually works. I would like to test this app to see how it actually works.

The portion after the comma is not a complete clause. Adding "to see" provides reasoning for your wanting to test the app.

I would like to test this app, and see how it actually works I would like to test this app and see how it actually works

I would first like to test how this app, how website actually works. I would first like to test how this website actually works.

I would like to test this app, to see how it actually works I would like to test this app, to see how it actually works