Adverb
ちゃくちゃく
その会社員は、いつも着々と仕事をこなしていた。
That office worker always got tasks done steadily.
プロジェクトは着々進んでいるようだね。
It seems like you're making steady progress on the project.
手はずは着々と整っていますよ。
The arrangements for it have been moving along steadily.
Something you wear, and wear again, is something you steadily wear. You do it so often, it's reliable. Also, if you wear something this much it's steadily wearing away too.
This word uses on'yomi readings of the kanji you learned, twice. You should be able to read this on your own.