Mockito NotaMockException

As you noted, act is not a mock, and therefore you cannot record behavior on it. You could use Mockito.spy to, well, spy (or partially mock) the act object so that you only record the behavior of secondMethod and execute the actual code for firstMethod.

Note, however, that matchers can't be used in doReturn calls regardles of how you're mocking or spying your object. A return value must be a concrete object.

class ActivityTest() {

  Activity act;

  @Before
  public void setup(){
     act = Mockito.spy(new Activity()); // Here!
  }

  @Test
  public void testFirstMethod(){

      Mockito.doReturn("someString").when(act).secondMethod();
      act.firstMethod();
      verify(act).secondMethod();
  }
} 

A slightly more elegant syntax allows you to use annotations instead of explicitly calling Mockito.spy, but it's a matter of taste really:

@RunWith(MockitoJUnitRunner.class)
class ActivityTest() {

  @Spy
  Activity act = new Activity();

  @Test
  public void testFirstMethod(){

      Mockito.doReturn("someString").when(act).secondMethod();
      act.firstMethod();
      verify(act).secondMethod();
  }
}