So this writing is inscribed into the floor stones of the Sanctuary of Artemis in Magnesia, near Söke, Turkey.
It caught my eye because the script is not standard Greek script, and i Iooked up ancient local variations of Greek script but it doesn't seem to match up with anything. Also, there are letters which look like the Cyrillic L (^) and others I just don't recognize at all. The final word looks too similar to "Artemi" to be a coincidence.
On top of all this, it seems more like graffiti than the other very defined writing to be found on the site. I'm really confused. Please help!
That's pretty normal Greek of the Hellenistic or Roman Period. It uses the typical post-Classical epigraphic script of the Ionic alphabet (what you're referring to as "standard" Greek), with nice serification on its letters and distinct letter forms. Υ is stemless, lunate Ε is present along with lunate Σ and three-bar Ε, Ω takes its normal Hellenistic form (identical to the later Byzantine minuscule ω), there are broken-crossbar Αs (with nice little serifs at the bottom of the crossbar), angular φ (potentially a bit odd, since this appears carved rather than scratched, but not especially unusual), Ρs that don't always join the curve and have a little stem at the top. All very normal stuff in Hellenistic koine inscriptions--this is certainly no epichoric ("local") alphabet or script, except in so far as that strictly speaking the Ionic alphabet is in origin epichoric. The only potentially odd thing about it is that lunate Ε is used alongside "normal" Ε, which is worthy of remark but not really especially strange.
What I cannot tell from your photos is...well, almost everything, actually. Which is not really your fault--the inscription is too large to be photographed from standing height, and you're not a trained epigraphist who could give me the information I need. I cannot tell how the lines work (or even if this is a single inscription), can't really see some of the letter forms clearly, do not know the spacing, can't tell how close the lines are to each other, etc. It clearly says εὐσεβῶν and φιλαρτεμιδῶν (I assume that's where the accent would go...). I cannot, on the basis of your pictures, make out the middle. If it is indeed the same inscription with no breaks then it should say something like εὐσεβῶν [.]αλαδαρ|ων φιλαρτεμιδῶν. Nothing comes to mind immediately for the word. Looks like some sort of cult or professional title, and I don't work on Greek or Hellenistic religion. Similarly, while the Philartemidae are easy enough to work out, I don't know of their attestation anywhere. I don't know what the publication of this inscription is and don't have the time right now to track it down, but it's a cultic inscription for a religious association of Artemis that I don't recognize, probably Roman Period, apparently proclaiming this space their property (it's genitive), which is not especially unusual (similar inscriptions exist on theater seats all over the Greek world).