1. go on adventure
go on an adventure - yes, common
2. go for adventure
- correct, but unlikely.
Why are you going? I am going for adventure. OR I am going for the adventure.
I am going for relaxation.
3. go to adventure
Correct, but unlikely
Why are you going. I am going to adventure. I am going to play. I am going to run.
4. go adventure
Correct, but unlikely.
Don't sit round the house all day watching TV. Go adventure.
Go play.
Go meet someone special.