How to keep fruits and vegetables fresh
As for fruits (including, for example, tomatoes), ethylene gas is released by fruits and causes them to ripen. You can buy "produce bags" that absorb ethylene gas, and slow the process of ripening. (Some fridges have drawers that absorb ethylene gas, but I doubt you feel like buying a new fridge.)

Update: See Vicky's answer and my comment for a couple links to examples of the "produce bags."  In addition to removing  the ethylene, the bags claim to also work by "breathing" and "reducing moisture formation."

For vegetables, it sort of depends. If you have root vegetables like carrots and beets, put them in water in the fridge (with the stems cut off), changing the water regularly. The roots will absorb the water (as they do in the ground) and thus stay quite turgid.