I'm an amateur photographer with a Canon Rebel XS, and I'm trying to figure out what type of lens I should buy to take better photos of my one year old when she is running around in our dimly lit house. I seem to find myself in a lot of situations where there are kids on the move in low light, and my 50mm f/1.8 II just isn't able to do it. Suggestions?
I don't think another lens is necessarily what you need, although since 50mm is a bit narrow indoors on a Rebel XS, you may want to consider something a bit wider. (The question Will a 35mm lens work for great indoor pictures of my kids? asks about Nikon but the answers will apply in general.) A Rebel-series camera has a sensor which is about 22mm wide, which means that the 50mm lens you have acts as a short telephoto. This is great for portraits of stationary subjects, but not so good for fast-moving small children indoors.
Something in the 28mm to 35mm range will be more "normal", a length which gives a convenient, natural perspective in many different circumstances. If you're feeling like it's often difficult to get everything you want in the frame with your current lens, this is something to explore.
For the low-light issue, though, the f/1.8 maximum aperture of your current lens is quite good. You can't go much faster without spending a ton of money. Image stabilization might help if your problem is that you can't stay still enough for longer shutter speeds, but I'm guessing the main problem with movement is in your subject, and nothing in the lens can help that. In fact, a fast lens (one with a wide aperture, like yours) is actually harder to use in this case, since your little one is apt to zoom right out of the narrow area of focus.
So, I think the main thing you need is more light. Since you probably don't want to add stadium floodlights to your house, take a look at Prime lens or flash: which upgrade will most improve baby photos? — with a focus on the "flash" aspect. A better flash provide a lot more illumination than the pop-up flash does, and more importantly, will allow you to bounce light off the ceiling, not just flash it forward, avoiding the harsh light and shadows often associated with amateur flash photography.
Canon's lower-mid-range flash, the Speedlite 430EX II, is a little pricey at $300 — but that's less than a nice lens might cost. It's also very well featured and able to grow with you as you upgrade the rest of your photography. You could also choose a compatible third-party option like the Metz 50 AF-1 ($200) or the more basic (all-automatic only) but very powerful Sigma EF610 DG ST ($150). Or, you can get a much-cheaper all-manual Yongnuo YN-560 ($70!), but that may not be what you want for in-the-moment kid photos.
I think that what you're trying to do is capture memorable images of your daughter. (Correct me if I'm wrong!). I don't think you're aiming for photographic perfection. As a result, many compromises - which may sound technically iffy - are fine for this kind of photography.
Bearing that in mind here are a few suggestions...
Don't be afraid to bump up the ISO
Even if the result is very noisy, if you get "the" shot, then nobody will notice the noise.
Slow down the subject
This works differently with different children, but giving them a toy (especially a cute, cuddly, photogenic toy) might make her stop moving for several seconds (!) - enough time for you to snap off a shot. The trick is not to let her see it until you give it to her. You could have a helper do this.
You could place a big cushion in her path and lie in wait. Negotiating the obstacle will present some interesting shots.
When you have no light, any light is good light!
Get a work lamp and put it in the room. I would point it at the ceiling to get good diffused / spread out light.
Set an appropriate white-balance (or shoot in raw).
Also, consider using the pop-up flash on your camera - perhaps dialled down (set the flash exposure compensation to -1 or even -1.5 stops).
Having a good lens is important for indoor shots: I recently reviewed a 35mm lens that I use with my Nikon Camera;
I would think a good 35mm lens for a Canon would produce similar results. It is also important to ensure that the ISO setting is appropriate for the light setting. Some people avoid ISO settings above say 600 because the image can get "noisy". However, modern cameras are better and I find I can get quite a good shot at ISO 1000+. It's important to experiment at different settings (ISO, f stop and shutter speed) to get a feel for the quality of images your camera can produce.