Usage patterns
本気で〈動詞〉to do (a verb) seriously/for real
He is studying seriously.
〈相手〉に本気になるto become genuinely serious about someone (fall for them)
He ended up falling seriously for her.
本気の〈事〉a genuine, serious (contest/attempt, etc.)
This is a serious contest.
Expressions
本気にするtake seriously; believe; fall for 本気を出すmake a serious effort; get serious; put one's heart into it