I have gotten some advice telling me to leave my beer in the primary fermenter for up to three weeks. I've also read that leaving beer mixed with yeast with no food left to eat leads to nasty nasty beer. What is correct?
It is correct but you can take hydrometer readings (for a consistent readying) once the krausen has dropped and either transfer to secondary or to bottles (style dependant) sooner to free up your primary.
I once left a nut brown ale in the primary for over two months (life issues kept me from bottling) with no ill effect. It sat at ferment temp for a month and the rest were at fridge temp (40F). Turned out to be one of the best beers I've brewed so far! Since then I've kept beers in the primary for at least three weeks before kegging or bottling; I have yet to use a secondary but then again I've only done nine batches since I started and have only made ales, porters and a hefe.