Without example pictures, it's difficult to tell exactly what the problem (or constellation of problems) is, but what you are describing could easily be the result of extreme underexposure of the film (more than 2 stops) without any compensation in development (that is, the film was developed for the normal time). Colour noise happens in film as well as in digital, and it is also more prevalent in shadows (where very few photons are hitting the film emulsion, so the chances of having large error compared to statistical expectations are larger) than in highlights or midtones. Digital amplification of a scanned image may make the problem somewhat more apparent than an old-fashioned optical/chemical print, but even a good old darkroom print of a severely underexposed colour negative will have both a lot of total visible grain (the equivalent of digital luminance noise) and a lot of discernible colour grain (chrominance noise).
It is true that "corner store" film processing usually results in relatively low-resolution scans by default (adequate for a reasonable 4x6" print, or an 8x10 if you're not too fussy), but it's not very likely that it's going to be at "feature phone" resolution and quality. It may appear so if you have a severely underexposed negative (likely, especially with a simple point-and-shoot, failure to set the proper exposure on a manually-adjustable camera, or a complete miss on the camera's ISO setting) or if you are using a surveillance-quality high-speed film (unlikely to happen without your prior knowledge). Knowing that the process involves digital scanning and printing may make you more inclined to see raster effects that aren't actually there, especially if the final image is comparable in quality to a noisy cell phone picture for other reasons.
You should, though, have the negatives, and ought to be able to at least attempt a re-scan. (If the images are severely underexposed, you wouldn't need to scan to see a problem; the negative image itself would be a barely-discernible, ghostly image on the negative rather than something that's just weirdly coloured and reversed in tonality.)