Before you do anything, check the "exposure compensation" settings on the camera (sorry, you may have to find a manual or ask someone here who knows the camera, I'm not familiar with Canons) and make sure it's set at zero - if it is non-zero, then that is likely the source of your pain. For colour negative film, it's pretty safe to set it to +0.5 or so and just leave it there. +2 or +3 can cause saturation and contrast loss, anything negative can cause loss of shadow details.
Further to what others have said, chromes (slides) will tell you if your camera is metering accurately or not, but if you're only using negative film then it shouldn't really matter that much, as long as you're not getting under-exposure. The slides are a good judge of exposure because you observe them directly and can tell if they're too bright/dark (look for completely-black or clear areas), whereas negatives go through a scanning and/or printing stage before you see the final image. Most exposure errors on negatives are hidden by the printing process, resulting only in some quality being lost, like what you're seeing here. Do your testing with chromes on low-contrast scenes, e.g. at dusk. Avoid bright sunlight and deep shadows while doing your test roll as it can confuse things, especially in Auto/P mode. You want a scene with as uniform a brightness as possible.
If the chromes tell you that the camera is dramatically over- or under-exposing, then there's your answer. Test a few frames both with the lens wide open (using A mode on the camera, the smallest f-number that can be selected) and with the camera at f/16. If the wide-open ones are exposed OK (ignore any softness) but the f/16 ones are overexposed, then that tells you that the aperture in the lens is sticky, which is a pretty common problem and for that lens basically means you bin it. Severe over-exposure of negative film can cause the sort of saturation and contrast problems you're seeing in your final images. If the effect is consistent with varying apertures, it means either your camera body is defective or you have accidentally adjusted it in error - see the exposure-compensation thing above.
If your problem is just sharpness and not exposure, then getting a Canon 50/1.8 is an excellent solution. It's a good thing to do anyway because it will teach you a lot about perspective (composition, etc) without you needing to know how to use the non-auto modes on the camera and it gives you a good lens to learn A mode on and try some shallow depth of field. Your Quantaray will be OK for 6x4" or so but will likely be soft for 8x10 and doesn't give you a lot of light.
There might be processing problems at your lab - with all the P&S users largely going digital (blasphemy here, but an option worth evaluating), the quantity of C41 (colour neg) film processed has declined dramatically so there are some poor places out there. Like others have said, if you want good skin colours and low-light ability, Portra 400NC is a good option. 400VC if you want brighter colours. Fuji Superia is very good value for money.
Getting a bounceable off-board flash is highly recommended if you're doing people shots. Avoiding the front-lit morgue-photo look is a huge improvement and generally more important than any choice of camera, lens or film you might make. You won't get that option with a P&S - good flashes are a very good reason to use a proper SLR.
On the matter of your P&S being better, that is entirely possible. For one, an SLR assumes you know a bit more about photography and leaves more up to you. Running it on Auto mode can definitely produce worse pictures, so it's worth investing a little time to learn about depth of field, apertures and shutter speed. The other thing is that the physical structure of a P&S (no big mirror box in the middle) means they can have simpler, sharper lenses cheaply. It is quite possible, even highly likely, that the lens on your P&S was a lot sharper than the cheap Quantaray lens you're using now. Of course, you can get much better lenses (and a variety of lenses for different purposes) for your SLR than you can for the P&S, but they cost.
