Lots of comments, didn't read through them all, so apologies if this has already been proposed. Many good options, but here's an easy one that worked for me. Do you have direct deposit at your employer? If so, open an online-only savings account at one if the many established banks. I have one through Barclay's. They typically have higher interest/growth rates than at a physical brick and mortar bank. The "downside" is they usually have fewer "features". This works in your favor! It makes it a little harder/longer to transfer money out of this account, but this acts as some imposed self-control.
Set up your direct deposit to divert a small amount of your paycheck to this account every pay cycle. 1%, 2%, whatever. Some amount that is so small you'll barely notice it missing. And here's the important part... DON'T CHECK THE ACCOUNT BALANCE. Forget it exists. Make a complex password you won't remember and hide the login information. Lock it in a safety deposit box, or give it to your mother or someone you trust implicitly so you can't access it readily. You will be surprised how quickly the balance grows.
Combine this with some of the other suggested techniques (budgeting, cutting out unnecessary expenses, etc) and as you decrease your expenses, increase the amount of your direct deposit into this account by an equivalent amount. You'll barely notice the "missing" money, and after even just a few months you'll be surprised at how quickly it accumulates. Not only does this build an emergency fund for you, but it's a learning opportunity. It teaches you by demonstrating how dramatically small changes in spending/saving habits can result in significant gains. It's a win-win-win.
Once you've got a fair bit of savings, and you trust your self-control, you can transfer some of these funds to a more traditional account that you can access at a moments notice, so you can use it immediately in case of true emergency (where waiting the day or three often required to complete a transfer from some online savings accounts is not an option).
Like I said, I can't guarantee this is an option for you, or would work for your situation. But when I was first trying to get my financial house in order while struggling to keep my head above water, it worked wonders for me. Best of luck. Keep at it, work hard, make sacrifices, and things will get better. I promise.