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How do I go about finding the correct projected co-ordinate system for an area of the planet I've never worked in before?

Is there a website or any such material anyone knows of that they could point me in the direction of?

The area in question is the Gardabani Municipality in the country of Georgia.

If possible I would like to get a CRS code for this which I can add to QGIS.

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Gardabani is 41° 28′ N, 45° 6′ E. The relevant UTM zone for places between 42° E and 48° E is UTM 38N, EPSG:32638. So this is a possible CRS you can use. However, depending an what you want to do, there are other, better, choices.

You can look up and search for CRS and EPSG codes here: https://epsg.io/32638 or https://spatialreference.org/ or https://georepository.com/crs_32638/WGS-84-UTM-zone-38N.html

Looking up at these sites for "Georgia" is a little bit tricky since there is a US state with the same name, so be sure to pick the "right" Georgia (or look for Armenia and than compare which of these is valid for Georgia as well).

If you look in georepository.com, you can select the place you're interested in on a map to get a list of CRS for this area on the globe:

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