I've only ever heard "To diet" or "To go on a diet", so between the two I'd pick the second.

1-to go on diet - this isn't correct English. 2 is correct.

Saying "go on diet" sounds like someone is trying to reduce weight. But "go on a diet" sounds like someone is going on a specific diet, can be to reduce or gain weight. That is my opinion about how they sound to me, but both are correct.