1. Is it a leafy green? If so, wash, dry, and if it comes in pieces bigger than spinach, remove any tough ribs and cut or tear into pieces smaller than an index card. Saute some garlic or onion in olive oil or butter, then add your green and cook over medium-low heat until wilted and soft. Taste. If bitter, add lemon juice or balsamic vinegar and give it another few minutes. If bland, add salt and pepper.
Alternatively, wash, dry, remove ribs, slice leaves into ribbons, and add to a vegetable soup. For tender leaves, add close to the end of cooking time. For tougher leaves, add earlier.
2. Is it a hard thing, like a root vegetable or a winter squash or a brussel sprout? Peel it if it has a tough skin, wash it if it doesn't, and chop into vaguely cubical bits a little under an inch on each side. Put the bits in a baking pan with a little olive oil and some salt and pepper, and roast at about 400 degrees F until tender and/or brown in spots.
3. If it is neither of the above, peel it if it's got a tough peel, cut it into bite-size pieces, and try a piece. If it's good raw, put it in a salad. (If it is in fact a fruit, an excellent salad can be made with spinach, nuts, and perhaps some additional fruits if you feel like being fancy. Balsamic vinegar and olive oil to dress.) If it's not good raw, you can roast it like a root vegetable (if it's squishy turn the oven down to maybe 300) or saute it in olive oil over medium-high heat until it looks tasty. If it doesn't want to soften, toss in a quarter-cup of water and put a lid on the pan so it can steam for a while. Then proceed as if it were a leafy green.
This will not always work (What happens if you roast 1-inch cubes of ginger? I don't know! Maybe I will try it and report back!), and it's never the most exciting presentation of the vegetable in question, but most of the time this procedure will at least tell you whether you like the thing and want to follow up finding more recipes or figuring out how to incorporate it into more complicated dishes.
If you have any questions or need more detail, hit up the comment box below.