Astro Panel (free; Android; shown above) is useful for checking when the next perfect observing night is coming up--and will even show you how long you've got before it gets too cloudy. If you live somewhere not famed for clear nights, this could help you really make the most of a less-than-ideal situation.
Ever wondered where the International Space Station is at any moment, or when it might next appear in the sky? Look no further than the ISS Tracker. This website provides the space station's current location, along with its rate of speed and from where on Earth the ISS can be seen.
NASA's own Spot the Station (above) will also help you track the ISS, and it lets you sign up for alerts that will notify you when the ISS is flying over your area.
Another great website to lose time to is Asterank, which has detailed information on over 600,000 asteroids. The site provides information on each asteroid's mass and composition, and works out the theoretical cost and benefit of mining each asteroid. While you may or may not be interested in asteroid mining, the site is great for data geeks, and it's fun to watch the asteroids orbit.
Taking photos of the sky
In general, if you want to take photos of what you observe, you're probably best off with a camera like a DSLR where you can adjust the shutter speed, use a telephoto lens, and so on. For example, you can get some impressive photos of the Moon if you turn up the shutter speed on your camera (1/500 to 1/1000 works) and use a telephoto lens. Long-exposure shots of stars can produce amazing images. Use a tripod for best results.
If you have a smartphone and a telescope, you can mount your phone to the telescope's lens and take photos that way. Numerous mounts exist, and you can make your own relatively cheaply.
In either case, experiment with your gear, and see what works best for you.
Other useful gear
Before you rush out, be sure you have a few items on hand, aside from your binoculars, such as a flashlight and a compass.
If you can get one, a planisphere--a circular star chart that can be altered by date and time--can help you know what you're looking it. Fortunately for the tech-minded stargazer, plenty of apps do a good job of replacing paper star charts.
But don't stop there--your local astronomy club can be a great place to learn more about stargazing, and to meet other astronomy fans. Many of these clubs present lectures led by professors and other figures within the field and hold all sorts of workshops.
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