To put out one's hand or extend one's arm in order to reach for something.
リモコンが届かないので、テーブルの上に手を伸ばした。
The remote was out of reach, so I reached out my hand across the table.
棚の一番上の箱に手を伸ばしたが、背が足りず届かなかった。
I reached for the top box on the shelf, but I wasn't tall enough to reach it.
赤ちゃんが抱っこをせがんだので、私は手を伸ばして抱き上げた。
The baby begged to be held, so I reached out and picked them up.
To try one's hand at something new; to reach out into a new field or activity (idiomatic).
彼は最近料理に手を伸ばして、休日にいろいろなレシピを試している。
Lately he's tried his hand at cooking and has been trying various recipes on his days off.
その会社は新市場に手を伸ばすことを決めた。
The company decided to try entering a new market.
趣味だった写真にプロとして手を伸ばすのは勇気がいる。
It takes courage to try turning a hobby like photography into a professional pursuit.