Yes, there is a rule. Whenever you can replace "to" by "in order to", you use "um zu" in German. See details here: https://yourdailygerman.com/use-of-zu-and-um-zu/.

Here are your sentences corrected.

Ich arbeite, um die Lebensmittel zu bezahlen.

Ich bin zur Apotheke (gegangen), um einige Medikamente zu kaufen.

"Das Lebensmittel" is wrong because that would mean that you work solely for buying one certain food item. "Einige Medizin" is weird because we normally use "Medizin" as uncountable noun. So, you either say "einige Medikamente" (countable) or "etwas Medizin" (uncountable).

Edit: Ok, after thinking about the phrasing again, I think it would be better to say "Ich arbeite, um das Essen zu bezahlen." Here, "das Essen" means food in general.

Thanks a lot Miriam. You are very kind. Deine Erklaerung is ganz klar.