Here in Oz, the aircraft must be instructed to contact Departures by the Tower. This can be either with the take-off clearance (if there are no spearation issues) , or once airborne (if Tower needs to monitor separation. Our MATS states that if Tower is providing vis sep, then the aircraft concerned must be on frequency).
Oz AIP states that in the absence of an instruction to transfer, the aircraft must remain on the Tower frequency, something few pilots seem to know.
Consequently, Tower controllers are getting into the bad habit of saying, 'remain this frequency airborne...' which is extraneous R/T, something no busy controller needs.
Although not standard, at my station the departure frequency is assigned at the end of the airway's clearance: 'Cleared to blah blah blah... departure frequency 118.4'.
As for when to instruct an airborne departing aircraft to contact Departures, well, I give up on the appropriate time. Some will be airborne on a climb gradient of 4.4 degrees (lot of terrain where I work), be dropping a wing as they pass 600FT in a 25 degree angle of bank turn and solicit "You want us on Departures"? Others will be wings level, passing 2500FT and when I instruct them, they'll tell me to standby and the Captain subsequently remark that I need to pick a more appropriate time to instruct them as they're 'quite busy'.