<p>A quick addendum: </p>
<p>I think that the timing of grad students joining groups depends on the university and field. In my university, it is common for incoming students to know already which group they wish to join, and they usually sign up as soon as allowed.</p>
<p>And another quick addendum:
When I say the costs of tuition and stipends, I mean the costs charged to grants of the professor who acts as the research adviser of the student(s) in question.</p>