So I've only just bought the game and was super excited to play it, only to find that the actual game play and cut scenes lag like crazy. I have turned down the graphics and lowered the fps but nothing has changed. Anyone else have this problem?

Post your specs.

Try this and report back: Turn off Anti-Aliasing in POSTPROCESSING settings.

You can also check this: Uncheck the "in game overlay" option in GOG Galaxy under settings, in case you are using Galaxy.