For full context: http://www3.nhk.or.jp/news/easy/k10011277371000/k10011277371000.html
The sentence in question: ローマ法王は、今までも「人間は広島や長崎から何も学んでいない」などと言って、核兵器に反対してきました。
My attempt at translation: "Concerning the roman pope, saying even until now 'mankind hasn't learned anything from hiroshima and nagasaki', he began resisting nuclear weapons."
「核兵器{かくへいき}に反対{はんたい}してきました」
Your translation of that is:
"he began resisting nuclear weapons"
I must say, unfortunately, that your translation is off.
「Verb in te-form + いく or くる」
always means "to do something continuously for an extended period of time".
It could never mean "to begin to (verb)" or "to begin (verb)+ing"
Thus, the phrase in question means:
"he has (always) been against nuclear arms"
The subsidiary verb 「いく」 is used to describe future actions and 「くる」, for past actions up to the present.
くる ⇒ きた ⇒ きました
The original uses 「きました」, the polite form of 「きた」.
