fortgehen
to leave
To leave a place; to go away.
Es wird langsam dunkel, wir sollten jetzt fortgehen.
It's slowly getting dark; we should leave now.

to leave / go out
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to leave
To leave a place; to go away.
Es wird langsam dunkel, wir sollten jetzt fortgehen.
It's slowly getting dark; we should leave now.

'Fortgehen' (to leave / go out) is a separable verb. The prefix 'fort-' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Wir gehen heute Abend nicht fort.
We are not going out this evening.
Since 'fortgehen' (to leave / go out) describes movement away from a place, the perfect tense is always formed with the auxiliary verb 'sein'.
Sie ist schon vor einer Stunde fortgegangen.
She left an hour ago.
The meaning 'to have fun' can often be recognized from the context. Typical signal words are time expressions like 'am Abend' (in the evening), 'am Wochenende' (on the weekend), or the mention of friends.
Lass uns am Freitag mit Freunden fortgehen.
Let's go out with friends on Friday.
to stay / remain
'bleiben' and 'fortgehen' can be opposites when the question is whether someone stays in a place or leaves it. 'bleiben' says: the person does not go away. 'fortgehen' says: the person goes away from this place. 'fortgehen' sounds a bit more general or more written than 'weggehen', but it is easy to understand. Also important: 'fortgehen' can sometimes mean 'ausgehen' in the evening, but in contrast to 'bleiben', the focus here is on leaving a place.

Nach dem Streit geht er fort und sagt kein Wort mehr.
After the argument, he leaves and doesn't say another word.
Sie bleibt in der Wohnung, bis der Handwerker kommt.
She stays in the apartment until the repairman comes.