If the space above this wall is attic space, then you've got a good chance of seeing which direction the joists run. As previously pointed out, if the joists run perpendicular to the wall, or you can see them bearing on top of the top plate of the wall you want to remove, then it's load bearing. If the joists run parallel to the wall, it's probably not load bearing. Since you're in the modd to do a little demolition, if the space above is finished second floor, then you can simply cut into the ceiling along the top of the wall to see where the joists run.
Now, if the wall is a bearing wall, that does not mean all is lost and it can't be removed. It just means that it probably can't be removed by you. You'll need a pro to temporarily support it, remove it, and replace it with a beam of adequate size to carry the load.
As you're new to the renovation game, it's best to have someone who has done this before take care of it for you. But watch what they do, if they don't mind. You won't learn by watching the TV shows. You'll learn by watching, and perhaps assisting in, the real thing.