This is a list of a couple of tips that can make your snapshots much better. This is not a list of tips for accomplished photographers.
This is the easiest way to make your pictures better. You know how it's rather difficult to figure out who's who in your pictures? That's because the person's face is making up approximately 2% of the total photo area. Get in real close, so that the face is the main thing in the picture.
Lot of the nice pictures isolate the subject by throwing the background out of focus. In order to do this, you need a fast aperture, something like f2 or so. Most point and shoot cameras don't have big aperature lenses. This allows smaller, crappier lenses to take decent photographs, but everything is in focus. It's a way of making sure the picture comes out right, even if the focus is off. Only bizarre point and shoots have fast aperature, and if you're going to pay that kind of money, you should just get a standard 35mm SLR.
If you have only shot with point and shoot cameras, borrow or rent a mid-prices SLR, and compare the photos you get from both cameras. The difference is usually night and day.
You can get crappy, but cheap zoom lenses, or really nice but really expensive zoom lenses. On the other hand, you can get a really good, cheap 50mm/f1.8 lens, which is what all SLRs came with before the zoom-craze. I bet I can take a better picture with the 50/1.8 than your 25-80/2.5-4 zoom.
Stick with print film. Modern print film is very forgiving, so that a couple of stops worth of over or under exposure can easily be compensated during printing. That means your camera could be slightly out of whack and still take good pictures. Not so with slide film. But buy the right film. Taking pictures in bright sunlight? (like sporting events?) - Buy 100 speed. Pictures in cloudy environments? - 400 speed. If you want to take pictures indoors without flash, 400 speed or 800 speed is your best bet. Stay away from 200 speed film. It's not much faster than 100, but your result will be worse.
(A note to the un-initiated - Higher number means you can take the photo with less light, but the grain will be larger. But these days, unless you get huge 8x10" enlargements done, it's hard to see grain with even 800 speed film...)
Personally, I stick with Fuji Superia 400 speed film. 400 speed is about right for well-lit rooms and a lens speed of about f2 and a hand-holdable shutter speed. This film reproduces bright colors nicely. It's a little too bright for skin tones though, so makes blemishes stand out. The film is usually affordable, like $12 for a 4 pack of 24-exposure rolls.
If you want no skin blemishes, a proper photo store will stock film for wedding photographers, which tones down the color improving skin tone. Ask for some kind of portrait film.
Expect to get about 2 good shots per roll of film. Even the best photographers waste film. This makes finding cheap film development important. I stick with supermarkets that have sales. Color print film is developed in big machines, so it shouldn't really matter who does the developing. If you get a negative with a photo you really like on it, get an enlargement done by a nice photo shop.