Zombie Apocalypse is a story-telling game. Most of the time, the story will be moved forward simply by telling what your character is doing.
Game mechanics come into play as soon as you attempt an action that is represented by a move. Usually, that just means to follow the instructions found in the move's description. This often involves rolling two six-sided dice (but see Advantage below) and adding a single Stat, e.g. roll +Sick means rolling 2d6 and adding your character's Sick Stat to the result. Usually, on 10 or more you get a plain success, while 7-9 means that you succeed, but with some strings attached. A miss is a roll of 6 or less, with less desirable results.
Sometimes a move gives somebody an Advantage, meaning that he gets to roll an additional die on his next roll. Only the two best dice count, you'll discard the worst one. Disadvantage works the same way, but this time only the two worst dice count.
If a move tells you that you've an Ongoing (dis-)advantage, you keep using three dice accordingly until the condition justifying the (dis-)advantage ceases to exist.
Trust is a special Stat used between the group's members. You can build up Trust with different moves, and you can use it to cry for help or to betray someone. See Basic moves for details.
Certain situations or moves allow your character to gain Experience (XP). Everytime the rules tell you so, mark one XP. As soon as you mark your fifth XP, clear them out and choose one of the improvements found in your playbook.