To begin to drink; to start drinking (can refer to starting to drink alcohol, medicine, or beverages habitually).
夕食でワインを飲み始めた。
They began drinking wine with dinner.
医者の指示でその薬を毎朝飲み始めた。
On the doctor's instruction, they began taking that medicine every morning.
パーティーでみんなが次々と飲み始めた。
At the party, everyone began to drink one after another.