Go to your local bike store (LBS) and ask them for a bike box.
Most of the time they will give you one for free or point you to their dumpster.
Find a packing supply store and buy a big roll of bubble wrap and some clear packaging tape.
Take the wheels, handlebars and pedals off the bike and wrap them separately.
You may also remove fenders if you want to make sure they don't get dented.
I usually ship the fenders, handlebars and pedals in a different box if it's a really nice bike.
Wrap everything in bubble wrap and tape it all up. Place bike in the bike box and then try to reduce the size of the box as small as you can. Fill extra space with wadded up newspaper or Styrofoam peanuts.
You can cut the corners of the box and fold the sides in to reduce the height of the box. In shipping, it's the box size that matters. A difference of a few inches can mean $40 to ship or $80.
Get on the FedEx website and use their shipping calculator to figure out shipping cost. You can play with box dimensions here to figure out pricing.
For whatever reason, I have found FedEx to be the cheapest of the major shippers here in my area. They are also easy to get to since they are in Kinkos stores.
Some bike stores will also package bikes for a fee....I have no idea what that fee is.
Throughout this process....take note of how many times you tell yourself you will never ship a bike again....ha ha!
Good luck!