はじめに (hajime ni) - (N4) - While はじめる (hajimeru) is a verb used to express the act of starting, はじめに (hajime ni) is an adverbial phrase used to indicate 'at first' or 'initially'.
開始する (kaishi suru) - (N4) - Both はじめる (hajimeru) and 開始する (kaishi suru) can be used to express the beginning of an action. However, はじめる (hajimeru) is more commonly used in daily conversation, while 開始する (kaishi suru) is more formal.
始まる (hajimaru) - (N4) - While はじめる (hajimeru) is a transitive verb used to express starting an action, 始まる (hajimaru) is an intransitive verb used to express the beginning of an event or a state.
始めて (hajimete) - (N4) - はじめる (hajimeru) expresses the act of starting or beginning, while 始めて (hajimete) is the te-form of 始める and is used to indicate the first time doing something.