It would help if you said which camera model you're currently using, since different compact cameras (even from the same era, manufacturer and price range) can have wildly different feature sets. That said, let me list a couple of options that you're fairly likely to have available.
As other answers have noted, you camera may have a "snow mode" that tries to measure the exposure correctly for scenes with a lot of white in the background. Mine, at least, also allows exposure adjustment in this mode, so you can adjust the results of the automatic exposure measurement if they're still darker (or brighter) than you'd like.
Another option is to shoot in P mode, which your camera almost certainly supports. This should allow exposure adjustment, and you should be able to mimic "sports mode" by choosing a reasonably high ISO setting. If your camera lets you adjust the aperture or shutter speed, you can also set a fast shutter speed and/or a wide aperture (which, for a given ISO and exposure, imply each other).
Your camera may also have an option to enable spot metering, which causes the camera to auto-expose the image based on just the spot in the middle of the image frame, rather than on the entire image. (My low-end Canon compact has it hidden behind the "menu" button, but it's there.) As noted in the other answers, though, this can lead to overexposed images when shooting a dark subject like a black dog. You may be able to get decent results by aiming the camera just off the subject, so that the metering spot falls partly in the snow and partly on the subject, but that can get quite tricky.
Finally, since you say your camera is a Canon, you might want to give CHDK a try. It's a free alternative firmware for Canon compact cameras that adds a bunch of features more commonly associated with high-end models or DSLRs, such as manual exposure controls and, notably, the ability to save RAW photos.
The reason you'd want to save your photos in RAW instead of JPEG format is that it lets you adjust the exposure (and many other things, like color saturation) on your computer after taking the shot. Of course, you could do this with JPEG files in Photoshop (or GIMP or any other image editor) too, but due to the lossy compression and limited dynamic range of the JPEG format, trying to brighten a JPEG file too much tends to reveal ugly artifacts and noise. A RAW image file preserves the full dynamic range of the camera sensor (which, even on low-end cameras, is a lot better than what JPEG can store) and does not apply any lossy compression, so it's a much better choice for post-processing.
(In fact, when I shoot snowy scenes with my DSLR, I always deliberately underexpose the shots so that the snow doesn't burn out. That way, I can later push up the exposure in UFRaw in "soft highlights" mode, and maybe apply a bit of manual curves correction, to get a nice snowy background instead of just ugly saturated white.)
The major down side of CHDK is that it can have a much steeper learning curve than your camera's normal user interface, and even just getting it set up can take some learning. That said, if you want to do stuff like shoot RAW without spending money on a new high-end camera, CHDK is definitely the way to go.
Ps. Even if you're shooting JPEG, you can still do quite a lot with curves adjustment. For example, here's what got with a few minutes in GIMP: