(1) After breakfast we went sightseeing
(2) After breakfast we went swimming.
(3) After breakfast we went to sightsee.
(4) After breakfast we went to swim.
(5) After breakfast we went for sightseeing.
(6) After breakfast we went for swimming.
(1) and (2) are correct and idiomatic. (3) is ungrammatical (there is no verb *to sightsee) and (5) and (6) unnatural to the point of unacceptability. (4) is basically acceptable, but better replaced by (2).
The ability to form a (gerundive) go -ing phrase, while generally possible for almost any kind of sporting activity (go surfing/skydiving/parasailing, etc.) is limited to some extent by convention, so that while we say go shopping, we do not, for example, say *go chatting. These phrase are therefore best treated by the learner essentially as fixed expressions.
Henceforth, please do not overload one post with many different questions, but try to post them individually (see guidelines for posting)!