die Bühne
stage
The raised platform in a theater or concert hall on which the performers appear.
Der Vorhang öffnete sich und die Schauspieler betraten die Bühne.
The curtain opened and the actors stepped onto the stage.

stage
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stage
The raised platform in a theater or concert hall on which the performers appear.
Der Vorhang öffnete sich und die Schauspieler betraten die Bühne.
The curtain opened and the actors stepped onto the stage.

With the preposition 'auf' (on/onto), 'Bühne' (stage) distinguishes between direction (accusative) and location (dative). You go 'auf die Bühne' (onto the stage), but you stand 'auf der Bühne' (on the stage).
Die Band geht gleich auf die Bühne.
The band is about to go on stage.
'Bühne' (stage) is a common component of compound words that refer to the theater. Examples are 'das Bühnenbild' (stage design), 'die Bühnentechnik' (stage technology), or 'der Bühnenauftritt' (stage appearance).
Das Bühnenbild für das neue Stück ist sehr minimalistisch.
The stage design for the new play is very minimalist.
When 'Bühne' (stage) means the professional field of theater, it is often used abstractly. Phrases like 'zur Bühne gehen' (to go into the theater as a career) or 'von der Bühne leben' (to make a living from the stage) point to this meaning.
Schon als Kind wollte er unbedingt zur Bühne.
Even as a child, he absolutely wanted to go into theater.
In a figurative sense, 'Bühne' (stage/platform) refers to a forum that creates public attention. Typical expressions are 'die politische Bühne' (the political stage) or 'die internationale Bühne' (the international stage).
Mit seiner Rede betrat er die politische Bühne.
With his speech, he entered the political stage.
The colloquial expression 'etwas über die Bühne bringen' (to pull something off) is a fixed idiom. It means to successfully get a task or project done or completed.
Die Organisation der Feier war stressig, aber wir haben es gut über die Bühne gebracht.
Organizing the celebration was stressful, but we pulled it off well.
theater
'das Theater' and 'die Bühne' both belong to the world of performances. 'Theater' usually means the building or the art form, so the whole thing. 'Bühne' is the place in the theater where actors, musicians, or dancers perform. The words can be confused because people perform on a stage in a theater. In practical terms: you go to the theater, but you stand on the stage.

Die Schauspielerin kommt auf die Bühne, sobald das Licht angeht.
The actress comes onto the stage as soon as the lights go on.
Am Abend geht er mit seiner Schwester ins Theater.
In the evening, he goes to the theater with his sister.