Not dreaming, rowanwitch... peanuts are actually a legume rather than a "nut". The sprouting seeds, if continue successfully, will grow into a fairly large bush. By that, I mean, one about 2 feet tall and maybe 2 feet in diameter.
Don't over water or overly fertilize the plants. The top of the soil should be fairly dry and only moist if you stick a finger down into it. Water only once a week, if that.
The plants will turn yellow when the peanuts are ready to harvest, in about 100 to 110 days, at least in Alabama, here in the U.S. where they're grown by the tons.
Pull the whole plant up when ready to harvest and pick off the peanuts and allow them to dry in a cool dark place for about a month, then roast.
If you're truly adventurous, you could try the Southern U.S. delicacy of Boiled Peanuts...
Clean them (still in the shell) from all dirt and loose shell fragments. Place them in a pot or and cover with water by at least rwo or three inches.. Add 1/2 cup salt (not iodized) per 1 gallon of water. Bring them to a rapid boil and turn down the heat to a slow, but roiling boil for 90 minutes or so. When the peanuts are firm but not crunchy, continue simmering (not boiling) for at least 2 hours (some of ours have taken 4 to 5 hours). When they reach the desired doneness, turn off the heat and let them cool on their own.
You can leave them in the salt water as long as you'd like, but they become more saltier the longer you leave them in.
They can be frozen or refrigerated, but... Warning!... they become highly addictive so we've never had any left over to freeze... Best of luck!