Usage patterns
年始の挨拶をするto give New Year's greetings
I gave New Year's greetings to our business partners.
年始回りをするto make the rounds visiting people to give New Year's greetings
My father makes New Year's visits to relatives every year.
年始に〈事〉at the beginning of the year, (something happens)
I set new goals at the beginning of the year.