In modern Japanese grammar, はじめる is always 始める.　It means to start something. はじまる is always 始まる too. It means something starts. We don't use 初 for verbs. はじめて has two ways of writing, 初めて and 始めて, but the former is an adverb and the latter is a variation of the verb 始める. ex: 日本語学習を始めて、初めてこの単語を見た。I came across this word for the first time since I started learning Japanese. Even native speakers often mistype it.