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If I want to say sthg like, I called him twice and every time the answering machine went off at the first ring. Is that Ok?
What I mean is that when I call him, the call goes to voice mail directly without hearing a tone.? how can I say that in English without using a lot of circumloquio. Went off etc...

I called him ten times this morning, but his telephone was switched to answering machine.
I called him ten times this morning, but his telephone was on answering machine.
I called him ten times this morning, but all I got was the answering machine.

Edit: not quite the same.

I called him ten times this morning, but each time, I instantly got the answering machine (instead of the ring tone).

Some more ...

I called him twice, and each time the phone rang, it went straight/right/immediately to the answering machine.
I called him twice, and each time the phone rang, I got his answering machine right away.

When do we use went off?

You don't, unless you are talking of bombs or fireworks. Or alarm clocks.

In the sense of triggering: bombs, guns, alarms (including fire alarms and alarm clocks), and suchlike. But they're responding either to a pre-set timed stimulus or an unintentional one. I would consider, "He pressed a button and the bomb went off" to be badly phrased.

Here the answering machine is responding immediately to a intentional action. Additionally, answering machines are often personified: e.g. "the answering machine picked up".