
sich begeben
to proceed
To go to a specific place or move there. This phrasing is elevated or formal.
Nach dem Abendessen begaben sich die Gäste in den Salon.
After dinner, the guests proceeded to the salon.
to proceed / happen / renounce
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to proceed
To go to a specific place or move there. This phrasing is elevated or formal.
Nach dem Abendessen begaben sich die Gäste in den Salon.
After dinner, the guests proceeded to the salon.

it came to pass
An outdated, literary expression used to introduce a story or event; it happened that.
Es begab sich aber zu der Zeit, dass ein Gebot von dem Kaiser Augustus ausging.
And it came to pass in those days that a decree went out from Caesar Augustus.

to renounce
To give something up; to relinquish a right or possession. This use is very formal and rare today.
Der Thronfolger musste sich seines Anspruchs auf den Thron begeben.
The heir to the throne had to renounce his claim to the throne.
In the meanings commonly used today, 'begeben' is almost always used reflexively. This means it needs a reflexive pronoun (mich, dich, sich, uns, euch) that matches the subject.
Ich begebe mich jetzt auf den Heimweg.
I am setting off for home now.
The phrase 'sich an einen Ort begeben' is a formal or literary way of saying 'an einen Ort gehen' (to go to a place). It is rarely used in normal everyday speech.
Der Präsident begab sich zu den wartenden Journalisten.
The president proceeded to the waiting journalists.
'Sich begeben' is often combined with prepositions that indicate direction. Common phrases are 'sich auf den Weg begeben' (to set off), 'sich in Gefahr begeben' (to put oneself in danger), or 'sich an die Arbeit begeben' (to get to work).
Nach einer kurzen Pause begab er sich wieder an die Arbeit.
After a short break, he got back to work.
The expression 'es begab sich, dass ...' (it came to pass that ...) is outdated and sounds like something from a fairy tale or an old story. It is used to introduce an event solemnly.
Und es begab sich, dass ein Wanderer des Weges kam.
And it came to pass that a wanderer came along the road.
In the rare meaning of 'auf etwas verzichten' (to give something up/renounce something), 'sich begeben' takes the genitive. The thing being given up is in the genitive.
Er musste sich seines Amtes begeben.
He had to relinquish his office.
to go
Gehen and sich begeben can both express purposeful movement to a place. Gehen is the neutral everyday choice, while sich begeben sounds elevated or formal and is always used reflexively. That is why sich begeben is particularly suitable in official instructions.

Nach der Pause begeben sich die Gäste bitte zurück in den Saal.
After the break, guests are kindly requested to return to the hall.
Nach der Pause gehen wir zurück in den Saal.
After the break, we are going back to the hall.