I've always been curious about this kind of photography; tried it a few times and failed miserably. Just what exactly do you focus on when shooting the stars? And what are optimal camera settings? Can anyone help me with this?
You focus on infinity. Pick the brightest star/planet and zoom in with live view to focus manually. Or use any very distant light.
For
shooting stars (a.k.a. meteors), you would ideally want large aperture (naturally) and wide focal length, on a tripod, also dark skies if you can find some. Your 5D3 or 1DX together with 16-35/2.8 @ 16/2.8 will be perfect. Then use the appropriate ISO that doesn't overexpose the sky in 10s, point your lens towards the sky after midnight and start shooting multiple 10s exposures until you get a meteor in the field. The difficult thing is that you don't know were or when they show up. With a remote, you can just let the camera run and do something else in the mean time. Alternatively, you can use bulb mode and stop an exposure early if you actually see a meteor during the exposure (to improve contrast).
On average you will find only one every 30 min, unfortunately, but it helps if you do it on a night with known high meteor activity (meteor 'shower') such as the trusty Perseids in August (then you might find up to 100 meteors/hour).
Edit: Oops, I read
shooting stars, not
shooting the stars 
Please disregard most of my previous text. For shooting stars you can use essentially the same settings, but you don't need to do multiple 10s exposures. Use as long exposure your night sky allows. If it's longer than a few 10s of seconds, you will start to see star trails (depends on focal length and direction).
