What does it take to get a good picture of objects in the night sky?

It depends on your goals and your budget.  Some of the brightest objects (Moon, Orion, North American, Eta Carinae, Andromeda) can be photographed with a normal camera in dark skies, but the results will not be as good as as those taken with dedicated equipment.  If you think you may be interested in astrophotography, start with what you have!  There are plenty of websites that will show you what you can do with whatever you already have.

Do I need a telescope?

Not necessarily, but you can generally get better results with a telescope.  Some objects appear larger than the moon, so magnification (big zoom lenses or telescopes) aren’t necessary for those targets.  Andromeda is roughly 5 times the relative size of the moon, Carina is about 4 times larger than the moon, the Pleaides about 3 times larger, Orion about double. The moon easy relatively easy to get pictures of, and shutter speeds can be fairly quick.  But everything else is quite dim, so you will need long exposures and/or multiple images which you could then “stack” in order to bring out the details. Most of my pictures were taken using a 400mm focal length, which will be enough “zoom” to get many of the beautiful things in the sky.  A good quality zoom lens can get you “close enough” to get a large number of extra-solar objects, but you will need to consider the “rule of 500” to determine your shutter speed to avoid star-trails (your target becomes streaked in your photo).  Since the Earth is rotating, longer exposure times will be susceptible to the effects of that rotation.  The rule of 500 is a very generic way to set your exposure - divide 500 by your focal length.  If you have a 50mm focal length, you can expect that a (500 / 50 = 10) 10-second exposure will not have star trails, whereas a 500mm lens could only have an open shutter for (500 / 500 = 1) one second before the image would show movement.

Do I need a special camera?

Short answer, no.  BUT, most consumer cameras have an IR-cut filter that results in more “normal” looking pictures, which is what most people want their cameras to give them.  The IR-cut filter allows the sensor to create images that duplicate what human eyes can see - we don’t see infrared, and generally don’t want our pictures to show that either.  In astrophotography, we usually want to capture as much information as possible - much of what we can see in the night sky is in the Hydrogen emission range of the light spectrum, and “normal” cameras will filter some of that out.  In most cases, the level of IR filtering will not severely degrade your pictures, and you can photograph all sorts of objects with “normal” cameras.  However, for very long exposures (minutes) and lots of images to be combined (hundreds), a dedicated astro-camera with cooling will give you the best results, because digital sensors create heat, which appears as noise - so a cooled sensor will have less noise affecting the images.  Here are a couple of good websites for more information on using normal, consumer cameras:

https://skiesandscopes.com/astro-modified-dslr/

https://clarkvision.com/articles/do_you_need_a_modified_camera_for_astrophotography/

Do I need a special mount for tracking?

As with my previous answers, maybe.  If you can capture the objects you want with faster shutter speeds, shorter exposures than the “rule of 500” would suggest, then no.  But most of the spectacular objects will need long exposure times, and for that, you will need to compensate for the Earth’s rotation.  When buying a tracking mount, you’ll need to take several variables into consideration - what type of objects you want to photograph, how heavy is your equipment, and what is your budget?   For deep space objects like nebula and galaxies,  with exposure times longer than the general “Rule of 500”, you will need to account for the Earth’s rotation.  Also, for planetary imaging, since you will need a long focal length to have any detail at all, a tracking mount will be needed to keep the planet in frame.  I tried taking pictures of the Jupiter/Saturn conjunction a few years ago, with my Nikon D3100 on a cheap tripod, a knock-off 150-600mm zoom lens and a very cheap doubler to give me a 1200mm focal length.  I could see several moons with both Jupiter and Saturn, but no details of either planet - just large white dots and smaller white dots - but I had to re-center the planets for every 3rd picture because they were moving so quickly in the frame.  A tracking mount just makes astrophotography much easier.  In addition to a tracking mount, which moves at a pre-determined speed, you can add a guide scope with it’s own camera, connected to monitoring software, and that can add an extra level of accuracy and account for (small) errors in your polar alignment or mount movement.

What sort of special processing is needed?

For most astro-imaging, you will want as much exposure time as possible…generally, the more the better.  There are limits to exposure time for each subframe, based on your camera, on light pollution, and atmospheric conditions.  For instance, my set-up will only allow up to a 300 second exposure.  With my light pollution - Class 5 to 6 on the Bortle scale, Suburban skies - I need to use shorter exposures.  When the atmosphere is turbulent, I need even shorter exposures.  So generally, I’m shooting 60-180 second exposures.  Most of the pictures you see here were created by stacking many dozens to hundreds of sub-exposures.  There is a good variety of programs, both free and paid, that can stack your images, and also allow you to do post-processing, which I will add to this site in the near future.

Do I need special software?

If you are taking long exposures and stacking numerous frames, you will want special software to get the most out of your data.  I started with a free program called Siril, and there are hundreds of tutorials that show how to use it.  I also use PixInsight, which also has YouTube videos galore.  Both programs are fairly complicated, and will require numerous tutorials and hands-on practice to become proficient, but the results are definitely worth it!  See my recommended links page for a couple of great channels to follow.