To touch or handle something (or someone) with one's hands; to feel by touch or to handle physically.
その絵には触らないでください。
Please don't touch that painting.
子どもに勝手に触らないでください。
Don't touch the child without permission.
布の柔らかさを触って確かめた。
I felt the fabric to check how soft it was.
古い本はページが弱いから、優しく触ってください。
Old books have fragile pages, so please touch them gently.
To touch on or get involved with an issue or topic; to approach or meddle in something (often used figuratively).
事件の核心に触るような質問は避けてください。
Please avoid asking questions that touch on the heart of the case.
その問題に深く触るのは危険だ。
It is dangerous to get deeply involved in that matter.
他人の私生活に不用意に触るのは失礼だ。
It's rude to meddle in other people's private lives without care.
To be harmful to or to have a negative effect on something; to hinder or irritate (often written with the kanji 障る).
夜更かしは体にさわるから、早く寝なさい。
Staying up late is bad for your health, so go to bed early.
その言い方は人の神経に触ることがあるので注意した方がいい。
That way of speaking can get on people's nerves, so you should be careful.