Welcome to the forum.
Firstly, there is nothing wrong with using the auto modes P, Av, & Tv. It's OK to use manual mode too, but if you are just adjusting manual mode to what the camera's meter says...then you are just going to get the same photo as the auto modes...so why not use the one that's easier?
One benefit of manual mode is that you can adjust the exposure away from the recommended value but you can also do that in the auto modes by use EC (exposure compensation).
The problems you are having are due to a few factors. Firstly, is a lack of light. We think that most indoor locations are well lit, but that's because our eyes adjust very easily. A camera can't be fooled though, and the lighting is probably rather dark compared to natural daylight.
The three things that control exposure are; aperture, shutter speed and ISO.
A larger aperture (lower F number) lets in more light but every lens has it's limits...and I'm guessing you have the 18-55mm lens, which is quite limited because it's maximum aperture isn't very big. That's one problem, you lens isn't ideal for low light photography.
So if you're limited by the aperture, the next thing to adjust is the shutter speed...which you have done. The problem is that the shutter speed controls how motion is captured. The slower the shutter speed, the more that motion (either by the camera or the subject) the more blur you will get. If your little guy is moving around, you probably want to keep a minimum shutter speed of 1/60, maybe even 1/90 or faster.
So if you combine that those shutter speeds with your limited maximum aperture....you just don't have enough exposure for the light you are in.
The next option is to turn up the ISO. This will allow you to use faster shutter speeds (or just get more exposure out of the speed you are using). So you might be OK if you crank it up to 800 or 1600 etc. The problem with this, is that higher ISO gives you digital noise. That is often better than blur, but it's up to you to decide.
Ideally, you could just add more light. Maybe in the form of flash or maybe just by turning on more lights or getting close to a window (during the day).
Another option would be to get a lens with a larger maximum aperture. For example, the Canon 50mm F1.8 has a max aperture of F1.8, which is much larger than F3.5 and a whole lot larger than F5.6.
Using the on-camera flash will probably solve your exposure problems, but the light from that flash is very flat and boring. That is one big reason why an accessory flash is nice to have, because you can tilt the flash up and bounce it off the walls or ceiling, giving you nice soft light, and plenty of light for your exposure. Not to mention that a flash burst is very fast, which will freeze the movement of your subject.