Elementary grammar books cover the difference between '赤ちゃんは歩き始めた' and '赤ちゃんは歩き出した'. However, there seem to be one more grammatically correct way to say 'to begin doing something', namely Vstem+になる. I wonder how that fits into picture. While '働きになります' probably says 'I will start working', and '働き始めます' says 'I am starting to work' I am unsure what is the difference between '働き始めました' and '働きになりました'.
Follow-up question: if I want to say 'He started thinking it is good', I should use 思うようだ about third person. So これがいいと思うようになりました is fine (and means 'he finally thinks it is good', like maybe I convinced him), and first version changes to これがいいと思う始めるようだ or 思うよう始めた?