Some great advice here, and I find it hard to pick my own personal favourite.
I'm going to say something which - I think - hasn't yet been said, for the sake of variety. Don't just throw away photos that you don't like immediately unless they're obviously out-of-focus, etc. Think. Is there a different way in which this photo can be used? I've managed to save a few by thinking up new Photoshop possibilities. Try it!
best advice i ever got was from a learned lesson, always take my camera with me, no matter how laborious it may seem, or cumbersome, because like always, the best shot comes out of nowhere. today i ignored the rule and was driving past a 7-11, a car had driven right thru the glass and into the store, there were cops and firefighters everywhere (no one was injured) but the lady beside me in her van paid zero attention to the scene right beside her, she just didnt see it, and SHE was the person i wish i could've gotten a shot of...all that chaos right beside her and she's staring straight ahead - i mean, what's going on in her life that can distract her from those kind of surroundings? she interested me, and i didnt get to capture that moment. ok the second best advice was to ALWAYS take spare batteries!
Though I do not strictly adhere to this, it does echo in my brain and has encouraged me to have it nearby in instances where I probably would not have otherwise...and so I have some great shots I never would have been able to capture otherwise.
yeah, your one about not going anywhere without a camera is in the top 5 for me, but i think #1 is:
don't shoot from where is normal. (5-6 feet up) everybody sees the earth that way, and it will almost always be boring (landscapes as an exception). for example, if there's a toucan sitting twenty feet away, and i'm standing straight up, shooting it, the picture will have no depth. the picture's background will be more boring than dick cheney when he doesn't have a gun. instead, lay on the ground and start shooting. you'll still have some ground in the background, but likeliness is that you'll have some out-of-focus trees, too. and that's what makes a wildlife or macro or portrait photo pop.
The "sunny f-16 rule" - when shooting a sunlit subject the exposure should be 1/(ISO setting) at f-16 or equivalent. I realise it was much more important for film, but even with digital you can miss the exposure for the first shot, and then its gone. I used it with Kodachrome, which is unforgiving, and got perfect exposures always! The other is "simplify!!!" anything that does not contribute to conveying the theme in your image is, by definition, distracting! BTW - when following this you must first determine what is the theme you are trying to convey, and that in itself is important for getting a better photo!
Make sure everything you see in the viewfinder is what you want included in the piece: don't shoot, planning from the get-go to crop the picture later. Unless it's a special case where you want, for example, a square piece. and apparently, there are square format cameras!