I'm a bit of a newbie with my Nikon D5200. I see the longest the shutter can stay open for is 30 sec on this camera.
How do I get it to stay open for longer? Do I need a remote?
While you can use the Bulb mode and press the shutter release button, the button presses are likely to cause unwanted camera shake, even on the sturdiest tripod. I use and recommend an intervalometer such as this: http://www.newegg.com/Product/Product.aspx?Item=9SIA04D09G0109 Notice that the shutter release time can be set for any time up to 99 hours, 59 minutes and 59 seconds.
Put it in Manual (M) mode and roll the wheel until it shows 'Bulb' on the display. Press and hold the shutter button for the desired length of time. You don't need a remote, but it helps to prevent camera shake to a great extent.
P.S. The manual is your friend.
Long exposures are one area where electronic sensors can't do what film did. Electronic sensors accumulate noise over time, in addition to any signals due to light hitting the sensor. This is why the designers of such cameras limit the maximum exposure time so something around 30 seconds. If you could hold the shutter open, and have the sensor therefore accumulate date, for 10 minutes, then you'd get mostly noise anyway.
The solution with electronics sensors is to take multiple shorter pictures. For example, twenty 30 second exposures back to back gathers pretty much the same information as one 10 minute exposure, except that the sensor noise is reset to zero 20 times during that exposure. You then combine the 20 pictures digitally later to get something like the long single exposure.
Many digital cameras have intervalometer capability built in. I don't remember whether the D5200 does, but I know higher end Nikons do. With that capability, you can set it up to take the 20 exposures in sequence automatically.
However, twenty 30 second exposures back to back still isn't the same thing as a single 10 minute exposure as film would record. The problem is that the digital sensor still has quantization noise. This means you can get the effect of motion over the 10 minutes, but not all the same low light gathering capability of film for the 10 minutes. As a simplistic example, let's say the digital sensor can only tell you light levels from 0 to 15. If in 30 seconds only .9 accumulates, this will be reported as 0 each time, and after averaging this for 20 exposures you still get 0.
If you are using long exposure to accumulate faint light, then you have one of the few cases where you should probably use film. There are ways to reduce noise in digital sensors so that they can accumulate light for longer times, but these cost a lot of money and infrastructure. For example, cooling such a sensor reduces one type of noise. When you're spending a billion dollars putting a telescope into orbit, you can afford to do this, but the price will be rather beyond reach for the average D5200 user.