Go Transitive or Intransitive

I'm a little confused by the verb 'go'. I know that it is intransitive. My issue is that I intuitively feel that it may have some transitive uses when used in the context of activities. For example:

Let's go fishing.

It follows the template 'subject verb object' so I would assume that in this context "go" is transitive.

Can someone please explain to me why my intuition is wrong?

Let's go fishing.

Yes, "go" is intransitive here, and "fishing" is a verb, not a noun.

It's essentially a catenative construction.

1st person inclusive let-imperatives contain the catenative verb let together with an NP object (here "us" reduced to 's) and a bare infinitival clause (here go fishing) as second complement.

In your particular example, the catenative verb go has the gerund-participial clause "fishing" as its catenative complement.