Shall we run a goroutine for every async request, even when they are spawned from each other?

I'm developing a web application in go and I know in http package, every request is run in a separate goroutine. Now, if the code inside this goroutine queries a database then wait and using db result calls a remote api to fetch some related data and son on and so forth, shall I run each of these calls in separate goroutine or the one provided by http is enough?

That depends on what you're doing.

Each HTTP request should be handled sequentially. That is to say, you shouldn't fire off a goroutine for handling the request itself:

func myHandler(w http.ResponseWriter, r *http.Request) {
    go func(w http.ResponseWriter, r *http.Request) {
        // There's no advantage to this
    }(w,r)
}

However, there are still times when goroutines make sense when handling an HTTP response. The two most common situations are probably:

1. You want to do something in parallel.

    func myHandler(w http.ResponseWriter, r *http.Request) {
        wg := &sync.WaitGroup{}
        wg.Add(2)
        go func() {
            defer wg.Done()
            /* query a remote API */
        }()
        go func() {
            defer wg.Done()
            /* query a database */
        }()
        wg.Wait()
        // finish handling the response
    }

2. You want to finish handling something after responding to the HTTP request, so that the web client doesn't have to wait.

    func myHandler(w http.ResponseWriter, r *http.Request) {
        // handle request
        w.Write( ... )
        go func() {
            // Log the request, and send an email
        }()
    }