Setting the intonation on a guitar it tedious work. It takes time and skill to do it. If you are a novice guitar player then do not try to do this yourself. Take the guitar in to the nearest music shop and have them fix it for you. It will cost money but its better than having a messed up guitar right off the bat.
OK the list of things you will need:1) Philips screwdriver 2) Chromatic tuner 3) Pick (do NOT use your finger as your picks because they are not as accurate as a pick and this is all about accuracy) 4) new strings (the reason behind the new strings is because they are sharper and, once again, more accurate than old, rusty strings) 5)1/16″ Allen wrench
The first step is getting the new strings in tune, as they are, without holding down the string at any fret (if you have recently new strings that is fine but if there is any rust showing on the strings then its a definite change in strings). Once you have it tuned one time go through again because the strings stretch as they are tightened and eventually loosen. once you are done going through and tuning the new strings (about 3-5 times through should do it) then we can start setting the intonation.
What you want to make sure of first is that your strings are in tune as open, meaning that they are not held down at any fret. Then once they are in tune on open go through every fret, one string at a time, and make sure there is no buzzing of the string when you pick it. If there is then you need to take the 1/16″ Allen wrench and make the string higher by tightening the two Allen screws in the bridge saddle(the things that the strings got through in the picture). be sure that the screws are screwed evenly or it will still make a buzzing sound even if its up all the way. you should do this for every string, one at a time. Also every time you change something on the bridge you have to tune it again. Now if you go through a string that you did not already set the action on and didn’t hear any buzzing then evenly lower the saddle until you hear buzzing. Then raise it back up until you don’t hear buzzing so you get the maximum action out of your guitar. Besure you check every fret on every string so that its the lowest it can get without it buzzing.
After doing all of that its finally time to set the intonation. what you want to do first is, after setting the action, once again make sure it’s in tune. After doing so, start with one string and hold it down at the 12th fret. not to tight but not too loose. The best way not to hold it to tight nor too loose is to slide from the 7th fret up to the 12th fret. once your holding it at the 12th fret pick the string and see if it is in tune. it should be int tune with the corresponding string, meaning that if your on the G string then the 12th fret should be in tune to G. If it is sharp then you need to lengthen the string. To do this you need to take the Philips screwdriver and tighten the bridge saddle for that string (in the picture, the screws would be on the bottom). If the tuner is showing that the string is flat on the 12th fret then you need to shorten the strings length by loosening the bridge saddle screw. Remember that every time you turn the screw you have to tune the string open again.
once you do all of these steps you guitar should be sounding and playing great. Good Luck!