click button to start new interface.

Have you taken any of the online training courses?  The functionality you're describing (as well as clear and correct terminology to describe these things) should be mostly covered in those classes.

At present, it's a little hard to discern just from what you've posted here what you're attempting to do, and what you may have already tried and been unable to do. 

In general, a interface with a button will be used on a User Input Task in a process, and a button on one task can very easily be configured to take the user (instantly) to any other subsequent task in the process flow, in which the interface associated to that task can show the user literally anything you want.  Again, the training classes cover a good portion of all of this.  Many of us are here to help you with specific questions you may be left with after that, however it's really hard for us to do the basic fundamental teaching, especially in this setting / format.

There are multiple ways to achieve this. One would be to have a button (have submit attribute as false), you can have a flag to render another interface when flag has a value and reset the value. You should find examples in the available patterns (In Appian Interface).

If both the interfaces you are referring to are developed and you won't wish to make any changes in either of the interfaces, then plug the 2nd interface after the 1st one in the process model (activity chained according to your need). With the click of the button, the form gets submitted, and the flow takes you to the 2nd interface.

Thanks

Pop-up window is not possible on button click but it will be helpful if you can share the exact requirement here. But as you want to navigate between 2 screens, you can use flag functionality. On this create a local variable and on click of the button set the value of the local variable to 1 or anything and then use showwhen functionality of if else functionality.

a!localVariables(
  local!show:true(),
  
  {
    if(
      local!show,
      {
        a!richTextDisplayField(
          value:a!richTextItem(
            text:"Clicked"
          )
        )
      },
      {
        a!richTextDisplayField(
          value:a!richTextItem(
            text:"Un-Clicked"
          )
        )
      }
    ),
    a!buttonLayout(
      a!buttonWidget(
        label:"Click",
        submit: false(),
        saveInto: {
          a!save(local!show,not(local!show))
        }
      )
    )
  }
)

If you want to use it in process model then please have a look on activity chaining concept.

The only way a button can initiate a pop-up is to have a recordAction added on the interface using a!recordActionField()