I'm assuming the G10 comes with DPP?
Elements comes with a Camera Raw plug-in. So the short answer is, yes, it should work with your G10.
Here's the deal with Adobe and their Camera Raw software: Depending on which camera you have, you'll have to get an updated version of the Camera Raw software (currently up to version 5.4) and that is tied to a current version of Photoshop, Photoshop Elements, Lightroom, etc.
So the first version of Camera Raw that supported your G10 was 5.2 (they are up to 5.4 with 5.5 in testing). Camera Raw 5.x requires Adobe Photoshop CS4 or later or Photoshop Elements 6 or later (or I think Lightroom 2). So you'd be covered for now shooting raw with the G10 you have.
Just be aware that some day when you upgrade to a newer camera, that will mean a required upgrade to a newer version of Camera Raw or the CR PSE plug-in, and
that CR version may require you to upgrade to a newer version of your Adobe software. Not a huge deal with Elements at less than $100, but this can get expensive fast with the "full" version of Photoshop or any of Adobe's Creative Suite bundles.
One other comment, Elements and Photoshop are really more about editing an individual photo. Programs like Lightroom, Aperture, etc. more about categorizing, culling, and editing large batches from a photoshoot. If you are fixing the white balance and pumping up the reds for all 75 shots you just took during a dive, Lightroom or Aperture might be a better tool for the job. (DPP now has some of these batch editing features, though not as user friendly as you and I might like.)
The good news is that there are trial versions of Elements, Photoshop, Lightroom, Aperture, etc. so you can try them all out to see which works for you before you buy, though there is a quite a learning curve with some of that software, esp. if you are new to the process of touching up pics digitally.
FWIW: I use Lightroom (and
DxO) for 98% of my work and occasionally pop over to Elements for that 2% that need extra help (usually fixing some localized section of a picture or removing some object that detracts from what I was trying to shoot).