Welcome to the forum.
Firstly, the shutter speed has no actual affect on the exposure from the flash. The flash is much faster than the shutter so you could use a shutter speed of 1/200 or 1/20...and it wouldn't change the exposure from the flash.
Now, the problem you are having is because of the way the camera is programed to act/meter in Av mode. When in Av mode, the camera will meter for the ambient light and try to expose for that...even if you turn the flash on. The camera thinks you only want the flash for fill light. However, because it's rather dark, you need the flash for more than fill...but because you are in Av mode...the camera doesn't know that.
The easy solution is to use P mode in this situation. I will probably set the shutter to 1/60 and use a larger aperture...which would probably not be enough light for an ambient exposure...so the flash will light up your subjects.
However, I would recommend using Manual mode in this situation. It's easier than it sounds. Firstly, set the aperture that you want. The camera will set the flash power to match your aperture choice (as long as it's in range).
Remember that the larger the aperture you use, the more range your flash will have, and the less power it will need to use. So F would be a good choice. A higher ISO also helps with flash range and power usage.
Then you have to set a shutter speed. You can use just about anything up to the camera's sync speed (1/200 on yours, I think). As mentioned, this won't affect the flash exposure...however, it will affect the ambient exposure. The longer the shutter speed, the more ambient exposure you will get. If there is enough ambient light, a long shutter speed may still cause some blur...so you need to be careful about that...but it's fairly easy to test out a few different shutter speeds to see what works and what you can get away with.
Lastly, you can adjust the level of flash exposure by changing the FEC (flash exposure compensation). You might want to vary it...anything from -1 to +1 or whatever works for you.
One last thing...you would be well off by never using the built-in flash again. Get yourself a hot-shoe flash like the Canon 430EX, which is much more powerful and allows you to bounce the flash etc.