What is cheap? Something like this is pretty inexpensive ($55) as far as mics go and can be powered by batteries so you don't need a field recorder with phantom power to use it. I haven't personally used it, but as far as sound goes things can add up in cost pretty fast. A battery powered shotgun mic looks like one of the easiest and cheapest options as it plugs directly into your camera and costs very little, the alternate setup will probably run you $300 to $600 in equipment if you go with the more budget conscious gear. You might want to do more research on other battery powered shotgun mics before you buy anything, I've had good luck with Audio-Technica gear before and what I linked you has a good user rating, but there might be better options for you.

I use an AT875R short shotgun mic if I am not borrowing someone elses nicer equiptment and I am happy about it's quality, at $160 it's still a pretty cheap mic, but it also requires phantom power which is yet another expense. I use this recorder with my iPod, at $250 it's on the cheaper side of things for field recorders with phantom power. Although if you want the option of phantom power and don't mind being teathered to power you could run your mic through something like this, which at $150 is cheaper then my recorder.

The nice thing about having a recorder that supports phantom power is it gives you a lot more flexibility on what mics you can use, if you know a friend who has nicer audio equipment it makes borrowing that more feasible. Also if you outgrow your cheap mic you can upgrade to something better. There are nicer battery powered mic options but most of the professional equipment uses phantom power.

If you don't send an audio output to your camera you will need to sync the sound with the video, so clapping in front of the camera should help in making that easier, or you could use a clapper board if you just feel like leaking money. But attaching an audio out to your camera should be easy enough.