Angular listen on subscription start of rxjs observable?

It depends on what RxJS version you're using. With RxJS < 7.3 you can use defer():

defer(() => {
  loadingFlag = true;
  return this.http.doyourrequest().pipe(
    finalize(() => loadingFlag = false),
  );
});

Since RxJS >= 7.3 you can use new event handlers for tap():

this.http.doyourrequest().pipe(
  tap({
    subscribe: () => loadingFlag = true,
    finalize: () => loadingFlag = false,
  }),
);

since every subscription to http observable causes new http call it is safe to set such flag outside of the pipe, at least it is how I do it.

getSomeData(){
  loadingFlag=true;
  return this.http.doyourrequest().pipe(finalize(()=>loadingFlag=false));
}

if you really want to go trough sub count(which is always down to 0 after every request due to finalizztion of http obs) check the implementation of refCount() and share() operators which internally counts the subscribers

Edit:

You can encapsulate setting of the flag using dummy observable as an entry point eg

getSomeData(){
  return of(null).pipe(
      tap(()=>loadingFlag=true),
      switchMapTo(this.http.doyourrequest()),
      finalize(()=>loadingFlag=false)
  )
}

Instead of setting some indicator as a side effect in your stream, try creating an observable that returns the state of your request.

readonly makeRequestSubject = new Subject<RequestParams>();
readonly request$ = makeRequestSubject.pipe(
  switchMap(params => this.doRequest(params).pipe(
    map(result => ({ params, result,  state: 'complete' })),
    catchError(error => ({ error, params, state: 'error' }))
    startWith({ params, state: 'loading' })
  ),
  startWith({ state: 'notstarted' }),
  shareReplay(1)
);
readonly isLoading$ = this.request.pipe(map(x => x === 'loading'), distinctUntilChanged());
readonly results$ = this.request.pipe(map(x => x === 'complete' ? x.results : []));

1. You could return an observable from a method and use it like an overweight promise, or you could have a single field that contains the stream, and all dependents could just subscribe and unsubscribe to it when they want. next() on makeRequestSubject gets called every time you need to make a new request.
2. The outer startWith should emit first. This is optional, but might be useful in some ui scenarios, such as showing a message with instructions if no requests have been made.
3. The request is initiated in a switchMap so succcessive calls will interrupt the prior ones if they're not already complete.
4. The inner startWith operator will emit first unless the request is instantaneous.
5. catchError illustrates another typical use case beyond settings a loading indicator.
6. You could subscribe to request$ directly, or use isLoading$ and results$**. I typically try to avoid creating extra observables, and just make sure the model on my component in a master stream matches up with my ui bindings.
7. Params were just included because they might be useful in the ui.