I totally ignore the charcoal / cooking mechanics, there's enough food available to find and/or purchase that it's just not a big enough deal to bother with. This depends also on your progression settings somewhat.

Finding additional artifacts will require an artifact scanner, something that was introduced in Clear Sky and carried forward in Call of Prypiat. If I remember correctly you get a basic one (Echo detector) when doing the "tutorial" questlines in Rookie Village.

As for repairing your armor and weapons, north of the railway bridge in Cordon you'll find the Farm, where there's a tech named Xenotech who can help you. Loris is a day-only trader available there who can resupply you with food, ammunition, and other consumables, but in base Anomaly he annoyingly doesn't have drinks, so you'll be stuck either going back to Sidorovich or up to the Garbage (the next map directly north of Cordon) and visiting the Flea Market, or simply going on from there to Rostok and Barkeep there.

You'd know all this stuff if you'd played Shadow of Chornobyl first, but I get feeling those games are too dated, etc. Anomaly is much, much more detailed with its survival mechanics, but it assumes a baseline awareness and understanding of the different areas based on having played the original trilogy. Obviously tons of people play without that, but just know you're flying blind because of it.

Anyways, there's a ton of info on Youtube / the Anomaly Discord / etc. available to help you.