Please don't be afraid of higher ISO settings. While it's true that ISO 6400 is a bit much for the 600D (and ISO 12800 is for emergencies only, like surveillance or "get the shot or else" photojournalism), ISO 1600 is perfectly OK on the 600D and ISO 3200 will clean up acceptably.

Remember: look at the picture, not at the pixels. It will make you a much happier photographer, and probably a much better one. With a lens having image stabilization and IS turned on, you ought to be able to hand-hold your camera (with practice) in most indoor conditions at ISO 1600 or 3200, and most human subjects (well, most grown-up human subjects, at least) should be able to keep still for the expected 1/8 to 1/30 shutter speed with a kit lens set to its maximum (lowest f-number) aperture if they try. Trying to use a lower ISO will mean that you need to use a tripod, and that you'd have to "freeze" your subjects long enough that even their small, involuntary movements are going to show up as image blur. A quarter of a second doesn't sound like much time, but it's hard to keep a person really still that long.

Set your camera on aperture priority, and use the widest aperture you can. Don't switch to a lower ISO unless the camera tells you that the shutter speed it's going to use is faster than 1/60s. Grain (noise) is a pain, but a faster shutter speed and the resulting sharper picture will result in a much better picture than something that is virtually noise-free but blurry. If you're hand-holding, don't close down your aperture unless you can do that while maintaining a high enough shutter speed (and, depending on the shot, you may want to close down the aperture before you even think about lowering the ISO so that you can get more of the scene in focus).

If you can bring more light, do it. Even a cheap LED "camping lantern", strategically placed, can make a huge difference (without affecting anyone's power usage). The light it produces will probably be very blue compared to the existing light, but you can either filter it (if proper photographic filters aren't available easily, you can use amber-coloured cellophane gift wrapping) or use the blue to artisic advantage. A plug-in work or utility lamp may be useful too if the electricity it uses isn't going to cause anyone financial hardship. A DSLR is a sensitive instrument, so you don't need anything like the truckload of location lighting we used to need in the film days; a 23-watt compact fluorescent bulb added to the light that's already there can make a huge difference to the picture.