antreten
to begin
To begin something, for example a new job, a journey, or an office.
Nächsten Montag trete ich meine neue Stelle in Hamburg an.
Next Monday I begin my new position in Hamburg.

to begin / to compete
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to begin
To begin something, for example a new job, a journey, or an office.
Nächsten Montag trete ich meine neue Stelle in Hamburg an.
Next Monday I begin my new position in Hamburg.

'Antreten' (to begin / to compete) is a separable verb. The prefix 'an' stands at the end of the sentence in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Sie tritt morgen ihre neue Arbeit an.
She begins her new job tomorrow.
The perfect tense of 'antreten' (to begin / to compete) is always formed with the auxiliary verb 'sein' (to be), not with 'haben' (to have). This applies to all meanings of the verb.
Er ist gestern seine lange Reise nach Australien angetreten.
He began his long journey to Australia yesterday.
The verb 'antreten' (to begin / to compete) is often used in a formal, official, or ceremonial context. It sounds more elevated than, for example, 'anfangen' (to start) or 'beginnen' (to begin).
Der neue Präsident ist sein Amt letzte Woche feierlich angetreten.
The new president formally took office last week.
In this meaning, 'antreten' (to begin) is often combined with nouns that describe a new stage of life or a new task. Typical expressions are: 'eine Stelle antreten' (to take up a position), 'eine Reise antreten' (to begin a journey), 'ein Amt antreten' (to take office), or 'den Dienst antreten' (to report for duty).
Nach dem Studium hat er eine Stelle als Ingenieur angetreten.
After university, he took up a position as an engineer.
When it comes to competitions, 'antreten' (to compete) is often used with prepositions such as 'bei' (at/in, for a tournament), 'zu' (for, for an exam), or 'gegen' (against, for an opponent).
Die deutsche Mannschaft ist im Finale gegen Argentinien angetreten.
The German team competed against Argentina in the final.
The expression 'ein Erbe antreten' (to inherit an estate) is a fixed collocation and means to take on an inheritance with all rights and obligations.
Nach dem Tod seiner Tante musste er ein unerwartetes Erbe antreten.
After his aunt's death, he had to inherit an unexpected estate.
to begin
'beginnen' and 'antreten' can both express the start of something. 'beginnen' is general and fits with many kinds of actions, for example a course, a job, or a conversation. 'antreten' is used more with certain fixed situations: 'eine Stelle antreten' (to start a job), 'eine Reise antreten' (to set off on a journey), 'ein Amt antreten' (to take up an office), or 'zu einer Prüfung antreten' (to sit an exam). It often sounds more official or more strongly tied to a particular role or obligation. Learners may confuse the words because 'antreten' sometimes simply means 'beginnen'. But you cannot use 'antreten' for every activity. If no fixed expression is meant, 'beginnen' is usually safer.

Ihr tretet nächste Woche eure neue Stelle im Krankenhaus an.
You start your new positions at the hospital next week.
Er beginnt morgen mit der Vorbereitung für die Reise.
He begins preparing for the trip tomorrow.