I shoot FX so the rules are a bit different, but you should do okay with that lens and a little experimenting for Milky Way and star shots. You should get a star guide with a seasonal dial to find what you're looking for. Look North this time of year to find the best clusters.
For stars your Tokina should be set on it's lowest focal length setting, 11mm and be mounted on a tripod. Shoot with your lens wide open and the best focus you can get, (at or near infinity). Either use a remote shutter release or use the cameras 2 second delay timer. You can either use the Bulb setting or 15-20-25-30 second settings, no longer than 30 seconds or you'll get star trails, (maybe shorter with DX format, they'll look oblong and smear). Some people shoot star trails on purpose with very long exposures and like the effect, a matter of taste I guess. Start with an ISO setting around 1,000 and work your way up to around 2,200 or so looking at your images with each shot until you find what works best with your format. Finally the darker the better, little or no moon light or light pollution. If there's a pleasant foreground you would like to add to the photo you can light paint it with a flashlight while waiting for your exposure. Aim the flashlight at the foreground and wave it around ''painting'' the entire object for several seconds, (try 7-10 seconds). FX and DX use different settings to achieve the same results here, I've given you FX format information here, but with some playing around you should get some good results with your DX camera.
Now, the moon. You won't need a tripod for this, and ignore your light meter, the moon is a bright object surrounded by darkness so metering does more harm than good. Best results are with a 300mm+ lens, your 11-20mm is nearly useless for any detail, (all you'll get is a tiny bright dot in the middle of a dark background). With at least a 200mm lens, shoot at f11-f16, shutter speed around 1/800 and ISO matching your lens speed or close to it, (use manual focus to get a clean shot). That's a good place to start with room for experimenting on your own, and clouds passing by can have an effect on your results. FX/DX doesn't matter as much here.