I'm just guessing with out a sample photo, but you might be low on light. I'm also assuming you get a decent picture in auto, possibly with flash. Try P mode without a flash. If its blurry, you don't have enough light. So try jacking up the iso until you get a non-blurry P mode picture. Keep that iso. I understand aperture is important for what you want to do, but in aperture priority, the camera will select any shutter speed it needs to make a good exposure. It may very well select a speed that is too slow for you to hand hold and you will get blurry pictures. For starting out, I recommend shutter priority. That way you choose the shutter speed and avoid the risk of the camera choosing too slow of a speed. Set it for a minimum speed appropriate for your lens and never go lower. I don't know what lens you have, but lets say don't go lower than 1/30 sec. As you raise the shutter speed, the camera will lower the aperture # which is what you want to blur the background. If you are low on light, you won't be able to raise the shutter speed very much before hitting your lens' maximum aperture (lowest f number). At that point your camera should indicate that its not happy with exposure. If you shoot anyway, you will get dark pictures.
In general, if the camera is not broke, shaky pictures means too low of a shutter speed. Dark pictures means not enough light for the settings chosen.