Removing the strings
First, detune all of the strings of your guitar. You can use a peg winder to loosen the strings more quickly if you like. Once you have removed detuned all of the strings of your guitar, remove the bridge pins. The end of a peg winder has a tool for removing the bridge pins. If you don't have a peg winder, DO NOT use plyers. You could damage your guitar.
Installing your new strings
Once you've pulled the bridge pins out, you should be able to remove the strings from your guitar. You can use a pair of wire cutters if you like.
Insert the ball end of the strings into the bridge. Place the bridge pin just on top of the ball of the string and push the bridge pin below in one motion so that when the ball of the string hits the bridge patch below it's tucked just under the bridge pin. If you let the ball hang too low, the string ball will wear on the inside of the bridge pad when you pull on the string and it will eventually need to be replaced.
Next take the rest of the string and slide it through the tuner. The lower strings only need to wrap around the post two or three times. Slide the string all the way through the post, then grab the string with your index finger and thumb. Bend the end of the string up and wrap once over the top. Now the rest will should wrap below the wind and below the string that sticks up. Keep tension on the string so that you get good tight winds.
Repeat this procedure with all of the guitar strings.
Once you get to the thinner metal strings, you need to wind the strings more in order to hold them in place. So be sure to give plenty of slack. Once you slide the string through the post, wrap the string around and start winding the string, providing constant pressure. It's important to always put pressure on the string by pulling it while winding. If you do not do this, the string could easily break while you are playing.
Now that you've installed all of the strings, bring the guitar to pitch, using a tuner. Then clip the strings on the headstock. After you clip the strings, bend the end of the string back on itself in a loop so you don't stab yourself with the string later.
And that's all there is to it.