I'm currently paying $67/month for storage of stuff I really don't need. I also have some stuff in my apartment that could go. What's the fastest/best way to unload it with the least amount of hassle?
Fastest and best aren't necessarily compatible in this case.
The fastest way to get rid of it would be to give it away -- many larger charities and churches will come to your place and haul stuff away if you're donating it. You'll get a tax deduction for the value of your donation.
If you'd rather sell stuff than donate it, that'll take more time. You can list things on craigslist, although in my experience at least half the people who say they're going to buy something and pick it up never actually show up and don't have the courtesy to call you to tell you they've changed their mind, so that can be a hassle.
My advice would be to give away all the things you don't think would stand a good chance of selling, and then trying to sell the things you think are worth the effort to sell. If you think you could make $50, $100, or more on an item, then it's probably worth selling. Otherwise just give it away.