Yeah, it depends.  When in doubt, throw it out.  It's not worth making yourself sick.

For veggies, I throw out anything that has mold on it.  Never eat anything with black mold growing on it, especially - that can really make you sick.  If it's slimy or dried up, I also toss it.  If the spinach is only slightly wilted, I will eat it cooked.  Most veggies I will try to eat within a week, but some things seem to last a bit longer (broccoli, carrots).  If it still looks and feels fresh, I will generally use it.

Composting is not too hard - we have an Earth Machine compost bin (really just an upside down bucket with a lid on the top), so we just throw in kitchen scraps, paper towels (ones used without soap or chemical cleaners) and napkins, grass clippings, leaves, etc.  We turn it occasionally.  If you have a spot for a small compost pile or bin, you should give it a try.  It really reduces guilt in throwing away food, and it's perfect for vegans since you can compost everything you eat (with the exception of oil, if you deep fry things).