How can i keep my area rug to lay flat and stop wrinkling up on my carpet?
Time and gravity is your solution. If you plan on having your rug in the same position for a while, then it is worth applying some pressure to the rug. Move some furniture, bookcases, weights. You need to weigh down the whole rug, not just the corners. Put extra weights on the corners though. You could even turn the rug upside down and put the weights on it for a while. The idea is to erase the rug's memory of being rolled up. Another tip is to hang the rug over a clothes line in the sun for a while. This will flatten it out. Make sure the color and pattern side is not facing up to the sun or it might fade.
I have a rug in my living room and I have a coffee table that's on top of it. Of course I have tile not carpet. Maybe you can go to dollar general or Walmart and buy a cheap table or something to keep on top of it. You can even search online and order a table!
you can always put heavy furniture on the edges of the rug to keep it down.
Staple or screw the four corners down. Staples will look better but screws will be easier to take out when cleaning is necessary.