
gemeinsam
together
As an adverb, it describes an action that is carried out together with other people.
Wir sind gemeinsam in den Urlaub gefahren.
We went on vacation together.
together / shared
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together
As an adverb, it describes an action that is carried out together with other people.
Wir sind gemeinsam in den Urlaub gefahren.
We went on vacation together.

shared / common
As an adjective, it describes something that belongs to, concerns, or is shared by several people or things.
Wir haben ein gemeinsames Interesse an klassischer Musik.
We have a shared interest in classical music.
The word 'gemeinsam' (together/shared) is closely related to the noun 'die Gemeinschaft' (the community). Both words share the idea that something connects a group of people or is shared by them.
Die Dorfgemeinschaft organisierte ein gemeinsames Fest.
The village community organized a shared festival.
'Gemeinsam' (together/shared) can stand as an adverb without an ending or be declined as an adjective before a noun. As an adverb, it describes how something is done. As an adjective, it describes what kind of thing something is.
Wir kochen gemeinsam. (Adverb)
We cook together. (Adverb)
When 'gemeinsam' (together) is used as an adverb, it often appears with verbs that describe a social activity. Typical combinations are: gemeinsam etwas unternehmen (to do something together), gemeinsam entscheiden (to decide together), gemeinsam arbeiten (to work together), or gemeinsam lachen (to laugh together).
Lasst uns am Wochenende gemeinsam etwas unternehmen.
Let's do something together this weekend.
As an adjective, 'gemeinsam' (shared/common) often describes abstract things that are shared. Common combinations are: gemeinsame Interessen (shared interests), ein gemeinsames Ziel (a shared goal), eine gemeinsame Zukunft (a shared future), or gemeinsame Freunde (mutual friends).
Trotz unserer Unterschiede haben wir ein gemeinsames Ziel.
Despite our differences, we have a shared goal.
together
For an action carried out with others, both zusammen and gemeinsam are often possible. Gemeinsam emphasizes the shared participation or intention more strongly, while zusammen can simply express that several people are doing something at the same time. Use gemeinsam for the shared nature of the action and zusammen for a neutral connection.

Wir planen gemeinsam den Einkauf für das Wochenende.
We are planning the shopping for the weekend together.
Wir gehen heute zusammen zum Markt.
We are going to the market together today.