TRACON Controller reported not issuing a clearance correctly and then forgetting to advise the relieving controller about the aircraft's release. Aircraft departed and turned into a higher MVA than was expected.

Narrative

While working a busy combined session; an aircraft called looking to get his IFR release from a satellite airport. A position was split off due to volume and once the arrival push calmed down; I reached back out to the aircraft to accommodate him. Once the position was split off; the responsibility for the airport in question should've went with it. Being that it was fresh on my mind after hearing the aircraft while I was busy; I didn't immediately realize that it was no longer my responsibility so I proceeded with reading his clearance. doing so; I kind of fumbled through the phraseology due to lack of practice and also did not issue runway heading to avoid terrain. Shortly after I was relieved and upon giving the briefing; I failed to brief the IFR release I issued. I did my 2 minute overlap and left. Shortly after leaving I realized I forgot to mention something and went back in the room to tell him. By then; the aircraft had taken off and turned immediately into a higher MVA. I should practice my Phraseology and create a way to remind myself to remember to brief someone else in a timely manner