
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.

to go out
To go out in order to have fun, especially in the evening to bars, clubs, or restaurants.
Wollen wir am Samstagabend zusammen fortgehen?
Do we want to go out together on Saturday evening?
'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 go out / run out / assume
Here, Ausgehen and fortgehen both refer to leaving the house for entertainment. According to the meanings given, there is no difference in usage in this area; both work for the same evening plan.

Wir gehen heute Abend fort und essen in einem Restaurant.
We’re going out tonight and eating at a restaurant.
Wir gehen heute Abend aus und essen in einem Restaurant.
We’re going out tonight and eating at a restaurant.