I'm a complete DSLR newbie and just bought a used Nikon D3200 on ebay. The thing is, every time I take photos with the 55-200 objective, I get these weird spots on photos. They are especially easy to see in darker photos. Could you please point me to what I should check to find the problem and/or how I could fix this?
That's pretty classic sensor dust. The way to find out for sure is to shoot a photo of a light subject (sky, wall, sheet of paper) at the longest focal length your lens offers with the focus set to infinity (disable autofocus) and the aperture at its narrowest (large number). You'll see the dust as dark spots with reasonably-well-defined edges. They won't be completely sharp because the dust is actually on a glass filter that sits directly in front of the sensor. This filter and the sensor are sometimes used interchangeably in this context.
The first thing to try is the built-in sensor cleaner in your D3200. You'll find instructions on how to operate it on page 180 of the user manual. Taking another test picture as described above will determine whether or not the dust is gone. You may have to run it a few times. In the same menu is a setting to determine when the camera cleans the sensor. I keep my bodies set to cleaning at both startup and shutdown, and that seems to help keep the dust down. (The first digital body I owned was a D1, which didn't have this feature, and that thing needed cleaning frequently.)
If the dust doesn't come off, you'll need to resort to other means. Page 182 of the manual outlines how to open the shutter, hold the camera upside down and use a blower to try and dislodge the dust. I recommend a high-quality blower like the Giottos Rocket Blaster, which is designed not to circulate dust. Do not use any kind of canned air product to do this.
If the dust still isn't gone, the last step is to physically clean the filter. There are a number of tutorials on how to do this available if you search for them. I don't recommend using the brush or pen methods and prefer wet cleaning with a wand custom cut from a plastic card, Pec*Pads and Eclipse cleaning solution. There's a lot of false mystique about cleaning sensors, but the fact is that they're not much more delicate than the front element of your lens. They are in closer quarters, and if you're not comfortable working on it, a repair shop will be able to do the job for you quickly and for not much money.
Buy an F6...
Digital cameras get so dirty and the sensors are so vulnerable to heat or bumps or vibration. The only problem I got to close to this common digital dirt was once I trapped a hair inside the camera back and every frame had a black hair across the tranny :(
You better get it cleaned or buy a new one, but then you will have to get it cleaned too. Hope it is not a case of a failed screen.