How to use a string like a variable in Visual Basic

As peterG mentioned, CallByName() can be used. This however requires that the var1 variable be declared as Public at Class(Form) level:

Public Class Form1

    Public var1 As String = "test1"

    Private Sub Button1_Click(sender As Object, e As EventArgs) Handles Button1.Click
        Dim newvar As String = "var1"
        Dim othervar As String = CallByName(Me, newvar, CallType.Get)
        MessageBox.Show(othervar)
    End Sub

End Class

If var1 is Private then you can use Reflection like this (it still must be a Class level variable):

Imports System.Reflection
Public Class Form1

    Private var1 As String = "test1"

    Private Sub Button1_Click(sender As Object, e As EventArgs) Handles Button1.Click
        Dim newvar As String = "var1"
        Dim FI As FieldInfo = Me.GetType.GetField(newvar, BindingFlags.Instance Or BindingFlags.Public Or BindingFlags.NonPublic)
        Dim othervar As String = FI.GetValue(Me)
        MessageBox.Show(othervar)
    End Sub

End Class

Note that if var1 is a local variable then no method exists to retrieve its value.

If you are a beginner with VB, you may find this site useful: Visual Basic Language Features.

If you want to set a string variable to the same string another variable holds, you just use the syntax like you have already:

Module Module1

Sub Main()
    Dim var1 As String = "test1"
    Dim otherVar As String = var1

    Console.WriteLine(var1) 'test1
    Console.WriteLine(otherVar) 'test1

    var1 = "test2"

    Console.WriteLine(var1) 'test2 
    Console.WriteLine(otherVar) 'test1 (notice this did not change) 


End Sub

End Module