Step 1 - Loosen the Strings
It's up to you whether you take all the strings off at once or do it one string at a time.
Step 2 - Remove the string and the Bridge Pins
You might find the bridge pins a little stiff - you can either try pushing the string in at the pin, which normally solves it, or in extreme cases put your hand in to the sound hole and push upwards from underneath.
Step 3 (Optional) - Clean your guitar
While all the strings are off it's a good opportunity to give it a good clean. Try to avoid furniture polish (if you have to use it, use it sparingly!) You can also clean the fretboard with some Lemon Oil to keep it clean and supple.
Step 4 - Put the new strings on
Put the new strings on in any order you like - I go from thickest to thinnest but other guitarists have their own methods! Place the ball end of the string into the hole as above, and put the bridge pin in over it. Pull the string through until the pin catches the ball, then push the pin firmly in so it sits how it did before you took it out.
Wind the string around the tuning peg once or twice before feeding what's left through the eye and pulling it tight. You can then tune the string up a little to put some tension on it.
Repeat this process with the rest of the strings.
Step 5 - Tune the new strings
Once the strings are on, start to tune them one by one. Don't tune them too quickly as the new strings will need to stretch. If you're at all unsure, don't bring them all up to tune at once. You can either do this by ear or use a tuner - I'd recommend the Rocket RT10BK which clips onto the headstock of the guitar. It's also chromatic so you're not limited to the standard EADGBE tuning.
Step 5 - Enjoy!
Once you've tuned the strings, enjoy the bright sound you'll only get from new strings. Properly cared for, a set of strings is good for around 3-4 months depending on how often you play them. To keep them in tip-top condition I recommend using a string lubricant like GHS Fast Fret before you play. This helps to clean the strings and will help them sound better for longer.