zusammen
together
Together with one or more other people; in company.
Wir wollen am Wochenende zusammen kochen.
We want to cook together on the weekend.

together
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together
Together with one or more other people; in company.
Wir wollen am Wochenende zusammen kochen.
We want to cook together on the weekend.

'Zusammen' (together) can serve as a separable prefix for many verbs and then means that the action is carried out together. Examples are 'zusammenarbeiten' (to work together), 'zusammenfassen' (to summarize), or 'zusammenleben' (to live together).
Können Sie die wichtigsten Punkte bitte zusammenfassen?
Could you please summarize the most important points?
In the meaning 'in total', 'zusammen' is almost always used in the context of bills, prices, or counting things. The question 'Zusammen oder getrennt?' (Together or separate?) is typical in restaurants.
Die Getränke kosten zusammen 12,50 Euro.
The drinks cost 12.50 euros in total.
The phrase 'zusammen sein' (to be together) means to be in a romantic relationship. It does not only describe being in the same place.
Wie lange seid ihr beiden eigentlich schon zusammen?
How long have you two actually been together?
with
'mit' and 'zusammen' can both express that people are not doing something alone. 'Mit' is a preposition and needs a noun or pronoun, for example 'mit Anna' (with Anna) or 'mit dir' (with you). 'Zusammen' is an adverb and is more closely connected to the verb: 'Wir lernen zusammen' (We study together). Often, you can even combine both words: 'Ich lerne zusammen mit Anna' (I study together with Anna). They can be confused because the meaning is similar. The practical rule is: if you directly name the other person, you often need 'mit'. If you only want to emphasize the shared action, 'zusammen' fits.

Nach dem Einkauf haben sie zusammen gekocht.
After shopping, they cooked together.
Wir üben morgen mit unserem Lehrer für die Prüfung.
Tomorrow we are practicing for the exam with our teacher.