Looks like you need to check the neck to see how flat it is when strings have been tuned up to pitch. I would suggest it has a bit of front bow where the gap under strings will be higher around middle of the neck. If so, loosen off the strings a little bit and when looking from headstock down to the body turn the trussrod 1/4 turn clockwise. Tension up, re-tune and check height. I have a 1m metal ruler that makes things easier as you can place it near the nut and also at end of the fretboard where that joins the body to help see where things are at. Keep repeating 1/4 turns at a time until it is pretty flat.

Next thing to check is action at 1st fret and suggest using a capo on 2nd or 3rd fret and see how much space there is between top of 1st fret and the underside of the string. Generally there should not be much more than the width of a business card or slightly less. If the space is more then it would appear that nut needs some work and generally easier to sand or file the bottom rather than hack in the string slots from above as that requires special nut files to do a proper job.

Once you have sorted these 2 parts time to re-tune and check action at 12th fret. If it looks higher than a couple of mm time to gradually lower the bridge and remember to re-tune up to pitch after each and every adjustment. Once you have got bridge height close to where you think it should be, time to play every note on every string to check for any buzzing which will highlight low or high frets. Hopefully there are none and also no buzzing strings anywhere. If so, time to move on to intonation which is adjusting the string length so that open, fretted at 12th, and 12th harmonic all read the same on an electronic tuner. Plenty of info on the internet about doing intonation but it is very simple once you identify whether to lengthen or shorten the string length to be in tune open/12th fret/12th harmonic.

That should keep you occupied for a couple of hours but well worth it once everything has been properly set up.

Cheers, Waz