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.

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 is outdated and sounds like something from a fairy tale or an old story.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 mean that someone moves to another place. 'gehen' is the normal everyday word and fits almost anytime someone leaves a place or goes or travels to a destination. 'sich begeben' means something similar, but sounds formal or elevated. You are more likely to hear it in announcements, official texts, or very polite language. When talking with friends, people normally say 'gehen'. 'sich begeben' fits when the situation sounds official or is phrased in a particularly matter-of-fact way.

Bitte begeben Sie sich nach der Anmeldung in den Warteraum.
After checking in, please proceed to the waiting room.
Wir gehen nach der Feier noch kurz zum Bahnhof.
After the party, we're going briefly to the train station.