Jill's the easier of the two characters (has two more slots to carry items), so probably good that you're playing as her.
For the first zombie, you can actually run back to the dining room, and Barry will take it out in a cutscene, saving you a few bullets. Otherwise, you just have to shoot the zombies over and over until they die.
There's a few advanced strategies as well, such as using the knife. You can shoot a zombie a few times until they fall over, then equip your knife and get a few slashes in before they get back up. You can also play around and start to learn how/when they lunge in order to get knife slashes in without them grabbing you.
Every once and a while, you'll randomly score a headshot, which will blow the zombie's head clean off with one bullet. Once you get a shotgun, you can do this much easier by waiting for the zombie to be in very close, then aiming up and pulling the trigger.
Whenever possible, try to keep zombies alive. If you kill every zombie, you're going to run out of bullets. Plus, you'll notice that most zombie bodies stay on the screen as you backtrack around, maybe remember where the "dead" ones are too...
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