It would vary from country to country; in the UK we don't have Class F airspace any more and Flight Information Officers (FISO in the UK; Flight Information Service Officer) only operate in Class G airspace.
I would guess the pilot should ask something like 'Request IFR clearance and climb to flight level XXX'.
Hopefully the flight plan was already activated with a DEP message.
What happens at some places (but not all) in the UK is the radio operator/FISO/ATC at the departure airfield will notify the ATCC that the flight is about to depart and get a transponder code and frequency to pass to the pilot before departure so he can do his own request or if the airfield is unmanned, there is usually a phone number available for the pilot to phone the ATCC just before takeoff.