The typical sequence at a large airport is:

1) listen to ATIS frequency... note pertinent info.
2) contact clearance delivery with ATIS code... get clearance and Departure freq.
3) contact ground get taxi clearance
4) contact tower when ready to go for TO clearance
5) on climbout tower has you contact departure in freq assigned by (2)
6) departure hands you off to center (if IFR or sometimes VFR with FF)

This sequence would be the same for all aircraft departing.