Are the (player) cards in your hand supposed to be face up or not?
Both ways are correct and dependent on who is playing the game and what their skill and preferences are:

When players are getting used to the game they are encouraged to play with their hands face-up to simplify the communication and memory aspect of the game.

To increase the importance of accurate communication and memory and to give everyone something to contribute to the group plan, the players can play with their hands face-down.

From the official rules, p.7


  PLAYER CARDS
  
  When playing the Introductory game (4 Epidemic cards), place your
  cards face up in front of you, for all players to see.
  
  
  Only Player cards count towards your hand limit. Your Role and your Reference cards are not part of your hand.
  
  
  When playing the Standard (5 Epidemics) or Heroic (6 Epidemics) games,
  keep your cards private, so everyone has information to contribute to
  play discussions.
  
  
  Experienced groups may decide to play with open hands, if desired, in these games.
  
  
  Players may freely examine either discard pile at any time