It seems like I've been talking alot about features and not much on story in relation to the main game. Obviously the mini game has almost no backstory, you're in a descending dungeon with no way back or up other than to go deeper down. That is about all you need if not more than you're used to in relation to a simple shooter. The main game I'm going to have to question you though if you want me to actually tell you all about the story or not.
Without spoiling anything I can give you a small idea of how the main story progresses. You start out in a cinematic that leads you to a place you're unfamiliar with, putting you in immediate danger in which you are saved by a small group of apparent military background people. They explain the situation to you while plotting something to use you. You build a certain level of trust with each of the characters, and at a point you essentially take over becoming the boss of this group. You then built up a new trust with them so that they will follow you in battle. Your objective becomes one of three which all lead to the same end. There's a few bosses, moral implications for some of your decisions, a few good twists, and you eventually solve a majority of the problems displayed in the game. The "zombies" are actually mutated clones and are non infectious. There's a few other groups which you help in the process. You take down the corporation responsible for the mutations and restore people back to normal. You may also return to where you started from or you may not choice is yours.
I couldn't be more vague I know. At least it's spoiler free. I could go into greater detail but won't. The actual story though the details are what draw you in and make it particularly good, so how can I go about this?