Binoculars come in many different sizes but they are all ‘classified’ using one very simple method. This defines how much magnification they give and also the diameter of the big (or objective) lenses at the front of the binoculars; for example 7×50 would yield a magnification of 7x with an objective lens diameter of 50mm.
One very important concept in astronomy is that magnification isn’t everything, of more importance is light gathering power and that comes from the size of main lens. Think of the lens as a funnel for light, funneling light over a bigger area into your eye. For binoculars a larger ‘second’ number means you will see fainter objects. The downside to this is that a larger set of lenses means the binoculars are heavier and harder to hold so over a certain size, you would need to consider getting a tripod to hold them.
The higher the magnification too, will also make for an unsteady image, magnification doesn’t just magnify what you are looking at but also magnifies any of your movements as you hold the binoculars.
A compromise between these two should be aimed for. I started off (and still have) some 7×50 binoculars and they work very nicely for astronomy. Mine only cost something like £20 and were bought from a second hand shop.
If you go much over 7×50 then you might want to think about attaching them to a standard photographer’s tripod. You can buy little adaptor blocks which do this job very well. All that said, it can be rather uncomfortable looking through binoculars on a tripod if you are looking overhead, unless you are a contortionist. A nice addition to a standard tripod is to get a parallelogram binocular mount. These generally bolt to a normal tripod but they take the binoculars away from the centre of the mount making overhead use much easier. They also have the added benefit of allowing you to adjust the height for other people to look through without altering where they point. Quite a neat piece of engineering.
Image Stabilization is a new technology for binoculars and adds a lot to the price but does work very nicely to steady the image, if you can afford it, then this is a very good option, although personally I would rather spend money on perhaps a better quality pair of binoculars than cheaper ones with stabilization.