I can tell from the doorway which green piece in a room earned its spot and which one got picked up because it was on sale near the register. The good ones have some depth to them, glass with a little cloudiness, ceramic with a glaze that catches light unevenly, stone with veining that never repeats itself twice. The sad ones glow like a traffic light under a lamp, flat and bright in the exact same shade all the way across, and you know before you touch it that it weighs nothing at all.
Where green decor goes wrong is usually a matching problem. Put it against a wall that's already green and the piece just disappears, or worse, fights the wall for attention and neither one wins. I'd also steer clear of lining up three or four green things in a row on one shelf, it stops reading as color and starts reading as a theme, which is a different thing entirely and not a flattering one. On sizing, use your hand. A piece that fits in your palm needs a small surface and close company, a side table or a narrow shelf, not the middle of a long console where it'll look lost and lonely. Something you need two hands to lift can hold its own on a larger table or a mantel, but measure the depth of that surface first, because a piece that overhangs the edge by even an inch will nag at you every time you walk past it. The common mistake is buying for the color alone and forgetting the object has to behave like furniture does, with weight and proportion, not just hue.
In an ordinary room, green decor sits easiest with warm wood, worn leather, brass that's gone a little dull, and linen that isn't trying too hard. It likes company that doesn't compete with it, cream walls, pale stone, soft lamplight rather than anything overhead and harsh. If you've got none of this in the house yet, start with one substantial piece rather than a handful of small ones, a glass vessel with some real heft or a stone bowl you can feel the cold of when you pick it up. One good piece will teach you what the color wants around it far better than five cheap ones scattered across the room ever could.
Pick it up before you buy it, every time. If it feels lighter than it looks, that's usually the whole story right there.