A few years ago I picked up a book in the library by David Bach called The Automatic Millionaire. I picked it up because it sounded like a simple book. It was an enjoyable read for me, and didn't require a bunch of knowledge about financial stuff. The book tries to get you formulate a plan for yourself that requires very little thought after it is put into place. It made a lot of sense to me then, and still makes a lot of sense to me now, so I figured it could help others as well.
The basic premise is to pay yourself first. You prioritize your savings, making automatic deductions from your checking account into your savings. Preferably, you set it up so the other account is at a different place than your primary account so that you don't see the money every time you log in to your main account.
Even $50/month will add up! Once you get a bit saved up, you will be even more motivated to continue.
The numbers that get thrown around for emergency savings are 3-6 months worth of expenses. That's a pretty decent amount of cash to keep liquid, or available in your bank account, but it makes a lot of sense. If you had that much set aside, you could deal with most things life throws your way, without affecting your day to day finances.