ATIS, Clearance delivery, Startup, Ground, Tower. It's that order I start with to obtain my startup clearance on a controlled field, even if departing VFR. And they will tell you what frequency to contact next.
From a pilots perspective, Clearance Delivery is typically a rather quiet and sedate frequency, meaning if you want a radio check, have any questions, there's something wrong with the flight plan, or you want a non-standard route, you can discuss it at length without impacting operations.