I've become interested in Korean dramas lately, so I've been looking into them.
I'm getting into Korean dramas these days, so I've been checking out different things about them.
ーI've become interested in Korean dramas lately, so I've been looking into them.
「最近韓流ドラマに興味を持ってきて、ドラマについて調べています」
to become interested in ... で「…に興味を抱くようになる」
to look into で「〜を調べる」
ーI'm getting into Korean dramas these days, so I've been checking out different things about them.
「最近韓流ドラマに興味を持つようになっていて、いろいろ調べています」
to get into で「興味を持つようになる」
to check out different thing で「いろいろ調べる」
ご参考まで!