There are many benefits to keeping your strings fresh. Better tone quality, better right and left hand finger response, wider range of dynamics possible and more range of tone to mention a few. Besides, your banjo looks and feels better. A banjo with clean, new strings practically begs to be played.
Change with the same gauge of strings as you used before. Your intonation may be off if you use a heavier or lighter set than you currently use. Most electric guitars come with extra-light or '9's'. Acoustic guitars tend to come equipped with light strings. You can experiment with other gauges but this might entail a setup of your guitar to accommodate the change in tension. A trip to you local guitar shop might best be advised.
Step 1: Loosen and remove old string - when re-stringing a mandolin, you can save yourself some later readjustments by changing only one string at a time.
Step 2: Un-hook the old string from the tailpiece.
Step 3: Place the loop end of the new string on the tailpiece.
Step 4: Here's the pro's trick to find the right string length to get the proper number of wraps around the tuner: hold the string like this with your left hand fingers - right at the tuner. Use your right hand to reach over and grasp the string at the fifth fret. Pull the string up into your palm, keeping the string between your index and middle fingers. Remember to let the string slide through your left fingers, and then hold the spot when the string looks like this. Make a 90 degree bend in the string where your left hand has marked the spot.
Step 5: Put the string through the tuner and start tuning it up - remember to keep good tension on the string as you tighten. This will help stretch the strings as well as keep it seated in the nut.