I don't know if I should put it in the refrigerator or the pantry. Does it go in the crisper?
Eggplant should be wrapped in plastic and refrigerated. It will last 5-7 days this way.
It can also be frozen 6-8 months with the following process:
You should store your eggplant in a cool part of your fridge, in a plastic bag or (preferably) wrapped in plastic wrap. It should last for a good week like this.
Alternately you can blanch and freeze it to store for 6-8 months however the texture will be different when you thaw it because the cell walls will have broken down some. If you go this route, I recommend this method (from stilltasty.com)