After trying a bunch of binoculars at Star Parties and the like, I have a pretty good feel for about what aperture and magnification I'd like in a new set binoculars. I'm an eyeglass wearer so a long eye relief would be nice, but what else should I be looking for in a good set of binos?
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Aside from the factors you've mentioned, nearly all binoculars you consider will be equivalent in terms of optics. Given that, you'll want to look for durability and comfort.
I'm not sure how common roof-prism binoculars are, but you'll probably want to avoid these for stargazing. Roof-prism binoculars have the objective lenses more-or-less in line with the eyepieces--they're more expensive because they need to be more tightly collimated (images in each eyepiece aligned) in the factory. The benefit is that they never need to be recollimated, but because they use reflective surfaces internally, they can reduce brightness by something like 15% (not exactly good for night-time use).
The more common kind are Porro-prism binoculars (where the objective lenses are not aligned with the eyepieces--they're wider apart than the eyepieces) which do not reduce brightness appreciably and are less expensive. They may occasionally require recollimation, however, but if you're not putting them through heavy use (if you're only using them for stargazing, you probably will never even need to refocus them) this shouldn't be an issue.
First, avoid the very cheapest binoculars. Binoculars typically get much rougher handling than most optics, so must be robust, able to survive being banged around a bit without losing alignment (collimation). Look for even dark green coatings, preferable multicoated (coating on all glass-air surfaces). Prisms should be BAK-4 glass, not cheaper BK-7; you can tell by looking at the exit pupils (BAK-4 has round pupils, BK-7 appear cut off). For astronomy, look for sharp images all across the field of view.
After owning and using many different binoculars, the size I find the most useful for astronomy is 10x50. The first number is magnification, the second aperture (diameter of front lenses). 50mm aperture is the minimum for astronomy. 10x is about the highest magnification which most people can hand-hold. Binoculars are at their best when hand-held.
Binoculars to avoid are those with zoom magnification (their optical quality is generally poor) and "universal focus" (means fixed focus at an average distance, making everything in the sky slightly out of focus).
Stick to major brands, particularly camera manufacturers like Canon, Nikon, and Pentax. Orion has a particularly complete and generally good quality line.