Umgang mit
handling of
The way in which one deals with, treats, or handles someone or something.
Der sichere Umgang mit Chemikalien ist im Labor oberstes Gebot.
Safe handling of chemicals is the top priority in the laboratory.

handling / interaction
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handling of
The way in which one deals with, treats, or handles someone or something.
Der sichere Umgang mit Chemikalien ist im Labor oberstes Gebot.
Safe handling of chemicals is the top priority in the laboratory.

'Der Umgang' (handling/interaction) is the noun for the separable verb 'umgehen' (to deal with/handle). The meaning of the noun corresponds directly to that of the verb: 'mit etwas umgehen' (to handle something) becomes 'der Umgang mit etwas' (the handling of something).
Man muss lernen, mit Stress umzugehen.
You have to learn to deal with stress.
In the most common meanings, 'Umgang' is almost always used with the preposition 'mit' + dative. This combination describes how one handles something or with whom one has contact.
Der Umgang mit den neuen Kollegen fällt ihm leicht.
Interacting with the new colleagues comes easily to him.
You can tell whether the handling of a thing or social contact with a person is meant from the word after the preposition 'mit'. 'Der Umgang mit dem Computer' (handling/use of the computer; thing) vs. 'Der Umgang mit den Kollegen' (interaction with colleagues; people).
Der tägliche Umgang mit Kunden erfordert viel Geduld.
Daily interaction with customers requires a lot of patience.
The expression 'im Umgang' describes a person's social behavior. You can say someone is 'freundlich im Umgang' (friendly to deal with) or 'schwierig im Umgang' (difficult to deal with).
Mein neuer Chef ist im Umgang sehr unkompliziert und direkt.
My new boss is very straightforward and direct to deal with.
The meaning 'überdachter Gang' (covered passageway) is a technical term from architecture. It is used only very rarely in everyday language and is usually limited to descriptions of historic buildings such as churches or towers.
Vom Umgang des Kirchturms hat man eine wunderbare Aussicht auf die Stadt.
From the gallery of the church tower, you have a wonderful view of the city.
society / company
'die Gesellschaft' and 'der Umgang' can both have to do with social contact. 'Gesellschaft' means more that you are together with other people or have a group around you. 'Umgang' means more the contact or relationship with certain people, that is, how often and in what way you deal with them. The words can be confused because both talk about being with people. 'Gesellschaft' fits better for the situation of being together; 'Umgang' fits better for the type of relationship or contact.

Der Umgang mit den neuen Kollegen ist freundlich und unkompliziert.
Interaction with the new colleagues is friendly and straightforward.
Nach dem Umzug hat Anna die Gesellschaft ihrer Nachbarn sehr genossen.
After moving, Anna really enjoyed the company of her neighbors.