게 나서다

step up, put yourself forward

An adverb in "-게" plus 나서다, "step out front, take it on yourself". Most lines here are captions marking someone volunteering: 자신 있게 나선 "stepped up confidently", 비장하게 나선 "stepped up grimly determined", 이렇게 나서서 "putting myself forward like this". It carries a hint of taking on something nobody asked you to, and it is also used of a country or group coming out forcefully. Note the last line is a different verb entirely — 맛이 강하게 나서 is 나다 "come out", about flavour.