First, buy a new set of strings. Always have a new set on hand before starting to change strings. Countless guitars have been abandoned in disgust because someone took their strings off and got stuck right there. Be prepared. An ounce of nickel-steel alloy is worth a pound of disgusted abandon.
Now... You have the guitar lying on the table before you. You have a new pack of strings. What else will you need? A pair of side-cutting pliers. Your local hardware should have them in stock. If not, have them order a pair for you, regardless of price. This is the tool the pros use to get those perfect-looking, precision-cut string ends. Take one of the old strings off the guitar. Depending on what kind of bridge your guitar has (Floyd Rose comes to mind), this may be all you get done today.

Assuming a non-tremolo bridge, unwind the string from the tuning peg. Pay attention to the direction you're turning. You want to LOOSEN the string. Over-tightening a string can lead to facial lacerations or worse. If you have safety glasses and heavy work gloves, now might be a good time to put them on. After the string is unwound from the tuning peg, pull it out the back of the bridge. There! Half the job is done (for one string).

Take a break. Don't tense up now. Get some more coffee.

Open the new pack of strings and locate the correct replacement for the string you just removed. Good! Uncoil it, run it through the bridge (from back to front) and up to the tuning peg (the empty one). Pull it one or two inches past the peg, put a kink in it, and cut it off 1/2" past the kink. This is where those side-cutting pliers really shine. Hook the kinked end of the string through the hole in the shaft of the tuning peg and wind it up until it's taut.

Caution: remember the facial lacerations.

If you have a guitar tuner or pitch pipe, use it now. Tune the new string to its approximate correct pitch. If no pitch source is available, the following method can be used:
• 1st string: when tuned, will slice sharp cheddar at 32º F
• 2nd string: will slice sharp cheddar at 45º F
• 3rd string: will slice sharp cheddar at 57º F
• 4th string: will slice medium cheddar at 68.3º F
• 5th string: will slice mild cheddar at 81º F
• 6th string: will slice velveeta at 92º F

Please keep in mind that the Cheese Method is only a last resort. The results will almost certainly be messy, and it's not that accurate, either. (Note: cheese can be removed from many surfaces with ordinary lighter fluid)

As you replace each string, tune to pitch using the appropriate tuner setting/ pitch pipe hole/cheese sample. Stretch all the new strings by vigorously bending notes and/or wedging large objects under them. If a string breaks during the stretching process, DO NOT ABANDON THE GUITAR! Set the instrument aside until you cool down. Have more coffee. Go back to the music store and buy a replacement for that string. Repeat the string changing/ stretching process on THAT STRING ONLY. Re-tune all strings, stretch them some more, re-tune, stretch some more, etc., etc., until they stop going out of tune when you stretch them. Clean all traces of cheese from the strings, fingerboard, tuning pegs, bridge, and guitar body. (Note: you may have to remove the strings to do this.) You're ready to play!