I found the answer, but rather than delete this question I thought I should leave it up to help others, since many people seem to be having issues with coupons and discount codes.
In Odoo 9.0c:
The first thing that needs to be done is to make a separate pricelist for your items that you want discounted. You could name it whatever you like, I chose "Holiday 20% Off". Do this in Sales > Pricelists > Pricelists > Create
After that's done, you need to add the item(s) you want discounted manually to the pricelist by going to Sales > Products and selecting each product you want discounted, then go to the "Sales" tab of the product and editing the item, adding the "Holiday 20% Off" (or your own name) to the pricelist, along with the discounted price (like $120, if the item's original price was $150). Also add in the start and end dates for each item, if there are any.
After you have all of your items added, go back to Sales > Pricelists > Website Pricelist > Create.
Under "Website" enter your website host (for me, I just typed "Website localhost"
Pricelist: Choose whichever pricelist you decided for your discount
Country Groups: Choose whichever is relevant for your website.
After you're done with that, test it out. You should be able to add the selected items to your cart, and then on the checkout page in the "coupon code" box, enter your discount code and click "Apply". You should see the item's price update immediately with the new price that you entered on the item's Sales tab.
If the "Coupon Code" box doesn't appear, go into your website editor and navigate to the shopping cart, then under the "Customize" menu, scroll down and check "Coupon Code".
I hope this helps at least one person, I spent several hours trying to get this to work and no forum anywhere seemed to have an answer.