御食い初め[okuizome] weaning ceremony
為初め[shisome] outset
為初める[shisomeru] to begin to do
仮初[karisome] temporary
業界初[gyoukaihatsu] the industry's first ...

月初め[tsukihajime] beginning of month
最初[saisho] beginning
初めに[hajimeni] to begin with (used as an introduction)
初めは[hajimeha] at first
手初めに[tehajimeni] at first

詣る[mairu] to go
参詣[sankei] temple or shrine visit
初詣[hatsumoude] first shrine visit of New Year
造詣の奥深さ[zoukeinookubukasa] deep knowledge
造詣[zoukei] deep knowledge

恵方詣[ehoumairi] New Year's visit to a shrine or temple which lies in a favorable (favourable) or lucky direction
神詣で[kamimoude] visiting a shrine
参詣人[sankeinin] worshiper
物詣[monomoude] visiting a temple
詣で[moude] pilgrimage