I'm a beginner photographer and am in the learning phase. The other day I was shooting in the A-mode, I needed a small DOF but the lighting was a bit dark. So when I took the pictures the shutter speed was too slow and they were all blurred. I tried increasing the exposure but this did not help either. How do we get enough light to the sensor in this scenario?
The main way you balance out light against your needs for the image is the "exposure triangle" of the three main settings: ISO, aperture, and shutter speed. You have to choose your priorities: noise (ISO), depth of field (aperture), or the possibility of blur (shutter speed).
Outside of these three settings, however, you do have two other options. You can use a tripod, which will allow you to use longer shutter speeds without camera shake blur (but which will still allow for subject motion blur), or if you really want a low iso, small aperture, and fast shutter speed then you have to add more light to the scene--typically with a flash.
You can try continuous light sources, but they generally have to be pretty powerful to accomplish what you want. A hotshoe flash, while it's more difficult to visualize than a continuous light, can deliver a LOT of light in a small battery-powered package. And while your built-in flash doesn't allow you to change the light quality/placement very well, and often gives that dead flat white look, lighting with a hotshoe flash and especially an off-camera hotshoe flash with modifiers is a whole 'nother game.
If you want to go this route, however, I'd recommend three things. First, be absolutely sure you've mastered ambient-only (non-flash) photography. Be comfortable shooting in M mode and knowing how to juggle stops between your iso, aperture, and shutter speed, because that's simple stuff compared to thinking about light and exposure when you throw in flash, too. Secondly, budget for a TTL/OEM flash if you can. Flash can transform your photography even more than a new lens; you should expect to spend as you would for a new lens--at least for your first/only, TTL-capable flash so you can use it both on and off camera with equal facility. Thirdly, read a bit before diving in. The two websites I recommend to anyone who wants to learn flash are Neil van Niekerk's Tangents website, and David Hobby's Strobist website, in that order (i.e., learn on-camera bouncing before you go all studio-lighting style with off-camera).
When in Aperture mode, keep an eye on the shutter-speed and learn what is acceptable for your lens. When it gets too low, raise the ISO.
Depending on your camera, you have options to do that. Most cameras have an Auto ISO mode and a few let you select a minimal shutter-speed. It's not great because that usually depends on the lens (the actual focal-length, so even on the same lens, depending where you zoom).
Some cameras have adaptive Auto ISO, so it considers the shutter-speed before raising the ISO.
Try the three primary parameters, Aperture, Shutter, and ISO. You ahve tried exposure and aperture, have you tried increasing ISO. That will allow more light into the sensor.
And since you say it's indoors with less light, try using a tripod or something to keep the camera still. Remember, irrespective of what all the tutorials say or experts speak, as a beginner you will learn only by the trial and error method. Keep trying and have fun.