Getting started:
1. Select a lens with a wide angle & lower f-stop capability. I used a 18-55mm lens that goes down to 3.5, although I likely would have used a wider angle lens that doesn't distort too much if I had one.
2. Set the camera and the lens to manual mode (don't forget to move the lens switch back to "A" when you're done.)
3. Set the shutter speed to 30 seconds (30".)
4. Set the f-stop to the lowest possible setting (on a Nikon hold the +/- button while scrolling the command dial/wheel)
5. If you have a remote, set release mode to remote.
6. Set ISO high at first, like 3200 or even as high as it can go. If photos look too exposed, step down ISO to reduce noise. Look in your camera's manual to see if it has a "long exposure noise reduction" or "noise reduction" setting.
7. Set camera on tripod, set lens to widest angle, and make sure to focus. Since you're focusing to infinity, things will likely look best turning the focus ring all of the way (or close to.) A cheat is to focus at the distance when the camera is still in automatic mode, although it's easy to focus to infinity in manual mode.)
8. Point at stars and shoot. Adjust shutter speed and/or ISO as necessary. A longer shutter speed allows in more light, but stars may look too large if they're elongated by 30 seconds of the Earth's rotation. A faster shutter speed takes in less light but results in less distortion. A higher ISO allows more light but also introduces noise. A smaller ISO results in less light but also less grainy noise. The darker the sky, the more options you'll have with which to play.
Tips:
*Besides getting away from city lights as much as possible, try to find a location with a low horizon (like a field) to the east. Residual sunlight lingers to the west, even after "official" sunset so having a clear view to the darker east will make for crisper photos.
*If you want to test out settings during the day (with easy access to a computer to look up questions) try going to a room in your house with no windows, like a laundry room or the garage. Although it isn't exactly the same, it will give you an idea if your camera can make dark scenes look light. If the test shots aren't working, try making sure the camera is zoomed out all of the way - it will take in more light that way.
*A new moon or a night with minimal moonlight is best for star gazing.
*Any patch of starry sky can look cool, but if you have a wide enough lens, the Milky Way can look amazing! It can be found from the southern to northeastern part of the sky.
*In the field, play with exposure time and ISO to find the right balance of light & noise for your setting. Even if you think something is working, try playing around just so you can view quality difference later. Also, if you're start before midnight, there may be a hint of sunset in the sky that your eye can't see but the camera can. It's possible that something that worked well at 11:00 PM won't look as nice at 12:30 AM so keep checking to see what works.
*Long exposures can drain batteries. Make sure your battery is fully charged before heading out. It's great to have a backup battery.
*Make sure you have a clean SD card before heading out. It's hard to really gauge success in such dark shoots so you don't want to delete any photos until you're back on a good computer monitor. Consider bringing a backup SD card if you're out on location to a dark sky.
*After reviewing your photos from your first outing on your PC, one easy way to see what settings worked for you is to right-click a favorite photo, select "properties", and select "details." Use these as a starting point for your next time out.
*If you don't have a tripod and want to know what to get, a totally basic one that tilts is all that's needed. Here's an example of a $12 on Amazon: http://www.amazon.com/Aluminum-Samsung-Panasonic-eCost-Microfiber/dp/B00J8RM35Q. In the meanwhile, try placing camera directly on the ground or table (on a towel or blanket) pointing straight up. See what happens!
*If you don't have a remote, here's something like my $8 remote from Amazon, although make sure to check model numbers, etc. first: http://www.amazon.com/ML-L3-Wireless-Remote-Control-Nikon/dp/B00M2PSPHG/.
*A colder night will result in slightly clearer photos as molecules in the atmosphere are slower. Warmer nights are much more comfortable, though, and are more likely to be warmer than the dew point.
*On a related note, if you lower the f/stop, increase the shutter settings, and point your camera exactly at the North Star, you can try for one of those photos with rings around a star! I forget how long the exposure time is for this, but it's pretty long.
Every camera, lens, and lighting situation is different, and sometimes the lighting situation differs at the same location as the sun and moon rise and set. Hopefully these tips will help you find a launching off point for some fun, new starry photos!