Movement of sperm to the egg
Once the sperm reaches the cervix, through what does the sperm swim
  through to reach the egg?


It swims through the cervical mucus. Human sperm swim in a straight path in cervical mucus.


  Is the uterus completely filled with a fluid that allows the movement
  of the sperm, or does the sperm only swim through the endometrium
  (uterine lining)?


Yes, the uterus is completely filled with uterine fluid. They are attacked by many leukocytes while being helped by various contractions. 

I would suggest reading the Suarez 2006 review. It has very nice information of all stages of sperm moment.