I would advise you head to your local library and see if you can get a copy of "Understanding Exposure" by Bryan Peterson. If they can't get a copy then nearly any good introductory photography book should cover the basic same content (that one I mentioned is just one I know and one often recommended by others as well). Another option is the Scot Kelby Digital Photography books 1 through to 4.
With the camera there are 3 basic settings; aperture, shutter speed and ISO. Those affect your photographic exposure and are the same for any camera; digital or film. Digital might have lots of bells and whistles, but those 3 settings are still all that affects the photographic exposure.
Shutter priority mode lets you control the ISO and the shutter speed; the camera then sets the aperture based on the meter reading when you go to take the photo. Aperture priority is when you control aperture and ISO and the camera sets the shutter speed based on the meter reading; and full manual is when you set all 3 (ISO, shutter speed and aperture) and use the needle on the meter reading to set the exposure yourself.
In the two priority modes there is something called "exposure compensation". What this does is tell the camera that when it sets its controlled setting it must under or over expose from the meter reading. This lets you have a bit more control as you will find that the meter can be fooled by certain scenes and lighting conditions.
Now as a starting point the recommended shutter speed is 1/focal length. So if you've a lens (zoom or prime) at 50mm then in theory 1/50sec should be the slowest shutter speed you can hand-hold at and get a good shot (assuming good posture and holding methodology). Of course many people find that they are a little over or below the guideline and in your case chances are you might need faster shutter speeds in order to get a sharp shot.
If you are getting underexposure then look at the aperture in the viewfinder - there is a chance that when you ahlf hold the shutter button down (metering the shot, but not taking it - ergo the camera is reading the light and setting the aperture) that the aperture is flashing on and off. This means that there isn't enough light in the scene to expose correctly and that the camera has already selected the widest aperture (smallest f number). In that case you can add light (flash, reflectors) or you can lower the shutter speed, or you can increase the ISO.
Note its daytime for me now, but you might be in a different time-zone and if its nighttime and you're indoors it can be surprising how dark it is for a camera indoors under normal house lights.
A tripod would certainly help (and you might find using the timer or a remote release helps as well); as said you can get cheap ones. My advice is to avoid the $15 tripods sold in shops as most are not well built; they work but not well. Instead head to
ebay or even a local auction sale room - tripods are one of those things that, kept right, will last a very long time; and old tripods might be heavier than some modern materials; but they are still very stable.