Can I ask for clarification on some of the responses?
Igor Belegradek and Paul are both very specific about which classes should be taken, and that "good grades" should be achieved. How does one define "good grades"? Does this mean all As? Mostly As with some Bs? Or perhaps As in undergraduate class with As or Bs in the graduate level classes taken?
(I believe Paul addressed the related question of what "good" GRE scores are.)
P.S. Based on my understanding, it would be more appropriate for this to be a comment on one of the answers but I can't do this without reputation points. So apologies if I am breaking the MO etiquette.
Edit: Another question: Does it make any sort of impact to have been an undergraduate grader and/or course assistant?