Powershot A70 Photo Quality

I find the photos I take with my A70 are quite "noisy" and the quality isn't that great. Even on the Superfine/Largest Size setting, the photos are only 180dpi.

I feel like I'm doing something wrong and it is really gnawing at me. Is it just the way it is?

There are two separate issues here.

The A70's Large/Superfine setting is 2048 x 1536 pixels. For an 8-1/2 x 11" print, that works out to 180 DPI. So that's what you can expect from a straight 8-1/2 x 11 print. The rule of thumb is about 240 dpi for optimal print quality, which would give you an 8 x 6 print. There are post-processing techniques and tools you can use to interpolate for larger prints (Genuine Fractals springs to mind). BTW - if you're using digital zoom, all that really does is blow up the center part of your image, so you're effectively using less than the full sensor, and letting the limited processing power of the camera try to interpolate for you - you'd be much better off disabling digital zoom and post-processing the image using the software that came with the camera.

Noise, random bright/dark speckles in the picture, will come from using the higher ISO settings. If you're using one of the "automatic" (Green box Auto, Sports, Portrait, Night, etc.) settings, you don't have any control over what ISO the camera sets. If you use one of the other (program, Av, Tv, Manual), modes you can control this. Above ISO 100 on the A70 you'll start seeing this type of noise. If you're referring just to a "blocky" picture, digital zoom, or over-enlarging a picture, can cause that.