(1) could be used in a situation like "The road ahead is blocked, so I need to go another way", i.e. take another route.
I can't think of a situation where you would want to use (2).
(3) may be possible as the result of substituting "go on" for "go" in (1). "on" emphasises the sense of forward progress. It is unlikely to work as "go" + "on another way".