Those symbols are called 'kanji', and a kanji for a new beginning would be, in literal terms,
新しい 始める
"new/fresh" - Atarashii. "beginning, start" - Hajimeru.
(of course there is no definitive kanji for this*, so hiragana is included)
I've also seen the term "新しい人生のはじめかた" used, "Atarashii jinsei no hajimekata"... It's also the title of a movie, but it means "Starting a new life" or "Getting started on a new life", along those lines
*The actual term for a "new beginning" would be like a kanji for reincarnation, which would originate in Buddhism and would not be used very commonly in modern times; most people would not know what it means.
新しい 始める
"new/fresh" - Atarashii. "beginning, start" - Hajimeru.
(of course there is no definitive kanji for this*, so hiragana is included)
I've also seen the term "新しい人生のはじめかた" used, "Atarashii jinsei no hajimekata"... It's also the title of a movie, but it means "Starting a new life" or "Getting started on a new life", along those lines
*The actual term for a "new beginning" would be like a kanji for reincarnation, which would originate in Buddhism and would not be used very commonly in modern times; most people would not know what it means.