"go for outing" is correct and can be used in written English.
For example, you might write, "My family and I decided to go for outing this weekend and had a lovely time."
When I was growing up in the Allgäu in the postwar period you did not travel much, and if you did go for an outing now and then as the "economic miracle" set in, it was by bus to the Tyrol, to Vorarlberg, or at most into Switzerland.
Go for an outing or activity together.
Go for class outings to boost intra-class bonding.
When she went for an outing to the cinema, she devised a system with the projectionist that if one of her mothers needed her, a message would flash up on screen telling her to go.
Children also want to be consulted about where to go for the annual school outing and believe that teachers should not lose their tempers.
The basic problem Google is trying to solve here probably feels familiar to everybody who has ever tried to get a group of more than two people to decide on where to go for dinner — or any other outing, really.
