beginning of the year; new year
Refers to the early part of the year, especially the New Year period. Often used in formal or business contexts.
I offer New Year's greetings.
年始はいつも実家に帰ります。
I always go back to my parents' home at the beginning of the year.
New Year's call; New Year's greetings
Specifically refers to the act of visiting people to offer New Year's greetings, or the greetings themselves. Often used in set phrases like 年始回り.
正月には親戚に年始に行く。
During New Year's, I go to visit relatives to offer New Year's greetings.
I gave New Year's greetings to our business partners.
My father makes New Year's visits to relatives every year.
I set new goals at the beginning of the year.
年始 refers to the beginning of the year, directly opposite in time to 年の暮れ, which refers to the end of the year.
年始 marks the beginning of the new year, whereas 暮れ marks the end of the old year.