Sound like you have a couple different things going on, try to isolate where the buzz is coming from. A buzz on a depressed string sounds like the string is touching or vibrating against a fret further down the fretboard, a high fret or the saddle is not high enough to create the proper angle for the remainder of the string to clear the frets, or the saddle is not tight in the bridge allowing buzzing in saddle/bridge connection. You should be able to see a depressed string touching another fret if that is buzzing. If you are comfortable with your nut height above the first fret generally around 1mm, then the saddle should be adjusted so the string height at the 12th fret is about 2.5mm above the fret with the strings open. That give enough angle for strings above the fretboard the whole length. Also make sure your fretboard is flat and not bowed up or down. If you are getting buzzing on an open string it could be the nut slots or the saddle. If you hold your finger on a string on the back of the nut without touching the string where it breaks over the crest and that solves the issue, then it may be your fret slots are too high or too low on the back side to the nut. Try holding a finger on the back side of the saddle as well to help isolate the issue. Also make sure your strings wind down the tuner post to create more angle in the string between the nut and the tuner. Hopefully one of these will help isolate your issue, but try to find where the buzz is generated. If everything else seems right, try a different type of string.
