Here's a nice and short video that demonstrates just the essentials of casting.  One reason that I chose this video is that the instructor makes a point of checking behind him to make certain it is safe to start the cast!

Remember, when you cast, you're throwing a sharp hook and tangling line at a really high speed, so this can be really hazardous if you are not careful.  Taking the time to make sure that your "back-cast" won't hook into something or SOMEONE will help to ensure that your time on the water is fun, safe, happy, and (relatively) tangle-free!

Papa's Tips

• Notice that the instructor does not hold the line against the pole with his index finger.  Instead, it is just draped over his index finger.  This will help to make sure that the line is released when intended, and that the cast will land where you want it to.
• In this video, the instructor manually flips the bail closed after he casts.  That's certainly okay to do, but virtually all of today's spinning reels have an "automatic" bail set on a spring mechanism.  Once the cast has landed on the water, you can just start turning the crank (handle), and the bail will close automatically.
• Conventionally, if you are right-handed, you will cast with your right hand, and crank (reel) with your left.  Most of the spinning reels manufactured today have a "reversible" handle.  If you are left-handed, you can unscrew the handle from the left side of the real, and screw it into the right-hand side of the real (after removing a screw in plug that seals that side -- just move the plug over to the left side, and screw it back in).  You will still follow the video instructions, but as if you are looking in a mirror: cast with your left, crank with your right.
• The casting motion is very similar to throwing a baseball.  Perhaps the trickiest part is to learn when to straighten your index finger to release the line that's draped over it.  I find it is helpful at first to "point" where I want the cast to land, but pretending that I'm throwing it to a person that's about 15 feet tall that's going to catch it.  It's silly, but helpful; give it a try!
• As you first start to reel in your cast (just after the first crank, which should close the bail for you), watch your line.  If it is loose/slack/blowing in the wind, it may be useful to raise your rod tip up slightly to get rid of some of the slack.  At the start of reeling in a slack line is where most "tangles" can form, so being mindful of this will help keep things tight and tangle-free!