From this answer:
Intent myIntent = new Intent(CurrentActivity.this, NextActivity.class);
CurrentActivity.this.startActivity(myIntent);
The first activity will still exist when you start the second one. It won't be "running" but it'll call the onPause() method if Activity A is still visible underneath Activity B. If Activity A is no longer visible at all, it'll call the onStop() method.
You said you don't want to use a Service, but if you want to still have some part of Activity A running while a new activity is shown, I would consider using a Service or some kind of Threading that gets started in either the onPause() or the onStop() method of the activity.