2 sentences

1. He suggested going to the party/on the party/ for the party.  ----is only "to the" correct?

 'On' is wrong. 

'For' gives a special meaning. eg

Person A: We have been invited to go on a 10 mile walk, followed by a party.

Person B: Let's just go for the party. (ie not the walk. The 'for' has a 'selecting/scheduling' kind of meaning.

'To' is by far the commonest choice.

2. You can prove her wrong.

   You can prove that she was wrong.

---are both of them OK? Both are OK. But the former does not necessarily relate to the past. 

eg It might mean 'You can prove that she is wrong'.

Best wishes, Clive