Hey JT, assuming no flash there are four important factors to balancing lighting:
1) Focal length of the lens
2) Lens aperature
3) Lens quality (generally how much glass and how well it's ground)
4) ISO setting
A few basics ... generally keeping everything else the same you will get need less exposure time for larger focal length (more zoomed) shot although the difference will probably not be enough to help you here.
It seems like you have everything else covered here except for aperature. Numerically aperature is the width the shutter opens relative to the focal length. The smaller the aperature number the wider the opening. You're probably already there in auto mode but you want to make sure the camera is using the smallest aperature number possible for this lens. To make sure you need to know your lens specs.
I'm assuming you have the 18-50 sigma lens that's rated for f3.5/5.6 aperature? This rating means you can get an aperature of f3.5 at 18mm and only f5.6 at 50mm. I'm familiar with Canon (I have a 5d) but not the rebel specifically. For mine I can turn the "on" switch past "on" and then when I push the shutter button part way to do a shutter check I can use the thumbwheel to adjust the aperature down and it will automaticaly adjust the shutter speed up to match. If that doesn't work for you you'll have to read your book to see how to set aperature manually.
Anyhow, if that's the right lens I would set the zoom as wide as possible (18mm) and then set the aperature as wide as possible (f3.5) and try some shots at the highest ISO. If you have any filters on the lens take them off.
If you still don't get fast enough exposures (1/30 minimum I'd say) it's time to throw some money at the problem. (Specifically your money.

) I don't mean to harsh on your lens but it's relatively inexpensive so there is probably not a lot of glass there. When I went from an cheap ($100ish) Canon 28-90 f4/5.6 lens to a not cheap ($1200ish) 24-70 f2.8 L-series lens my exposure times were half or less under the same conditions with the better glass.
Unless you're going to be doing a lot of photography a lens like that is probably not a great investment. If you just want a lens to try things out there is a relatively cheap prime lens (i.e. fixed focal length) that may be good for you:
Canon | Normal EF 50mm f/1.8 II Autofocus Lens | 2514A002 | B&H
It's a 50 mm lens with a max f1.8 aperature for ~$75.00. Aperature refers to the diameter of the opening and the amount of light let in is proportional to the area of the opening. Therefore the amount of light is proportional to the square of the aperature. So if I have your current lens right and you have a max aperature of f5.6 at 50mm this lens would let in (5.6)^2/(1.8^2) = 9.7 times more light. (i.e the shutter speed would be almost 10 times faster for the same shot) I don't know if 50mm would be too far zoomed in for your needs but you can check with your current lens.
One final word about aperature is that it affects the "depth of field" of your shot. The wider the aperature the blurrier objects closer and farther from your focus point will be. Here's a sample shot at high (f2.

aperature shot close to the subject ... notice how blurry the background is.
Often this is desired as an artistic effect but it may be unwanted for you. Being farther away from the subject will make the foreground and background less blurry.
Good luck and I hope this helps and wasn't too confusing !!