If you want to record your lesson for future reference (and not to put it on YouTube), start with asking your instructor for permission. If it's just for private use I can't see a reason for objection, but if you intend to put it on YouTube, the instructor may have reservations.
If you've got permission, then make sure you position the GoPro so that you can see yourself moving the controls, plus the most important instruments. So probably a little ways behind the instructors head, pointing diagonally forward. Don't try to capture the inside and outside simultaneously: The contrast between those is typically too high to get a decent view of both simultaneously.
In order to mount the GoPro, use the suction mount. Works fine: I have never had a properly attached suction mount fall off, even when doing aerobatics.
Depending on the noise level in the aircraft you may just be able to get half-decent audio without any adapter. If you set the comms panel to Speaker, then it may even include ATC. Try it out. If it doesn't work (too much engine noise), you'll need a Y-adapter that plugs in a headset socket, and has a socket for your headset plug, plus a lead that goes to the GoPro. Something like this:
PA-80 Camcorder Adapter | 42122 - Note that the older GoPros use a 3.5mm jack for audio-in, but newer GoPros do this via the USB socket. Adapters for that are available. When routing the cable, make sure it can't foul the controls.
If you intend to put things on YouTube, it gets a lot more complicated. You need to start with the question: What story do you want to tell to the audience? You will then need to collect hours worth of video, shot from different angles, and edit that down to just a few minutes, if you don't want your audience to be bored out of their skull. This requires multiple flights, and requires you to think about the recording during the flight. Not a good idea during a flying lesson, as your attention should be elsewhere.