I use a couple of different techniques. The easiest is hardy veg. Sweet potatoes and onions, for example, last a really long time (and don't need to be stored in the fridge at all). Carrots and celery are good for this category too.
Then there's "live storage" - veg that is cut from a stalk can often be kept fresh by putting it's root or stalk in water. That's things like celery, leeks, chives, asparagus, even kale. For the stalk ones, just cut the tip off the root to give it a fresh way to drink the water. put a 1/2 inch of water in a cup and "plant" the veg in there. Replace the water every day (or the bottom may start to rot). I've had asparagus last 2 weeks this way, and chives will just keep growing (i just cut some that were 2 feet tall). This is a bit harder to remember and only works for some veg, but is really effective.
You can also check out various options for the best way to store things. For example, i've found that rinsing salad greens and storing them wrapped in a slightly damp hand towel (or paper towels) keeps them good for a week or two (versus a day or two in a plastic bag from the store). That's the only thing i've got off hand, but i found that method by doing a google search for storing veg that came up with a lot more.
The least good, but still viable, option is to buy less-fresh veg. If you can buy, say, a great "today" tomato, as well as two that are close to ripe but not quite there, and another two that don't look that hot, you may have a tomato for every day of the week if you leave them by your window to ripen.
There is also the option of progressing through the week from awesome fresh salads to more cooked-veg-based meals, because there's a big gap between "i don't want to eat that raw" and "rotten," where you can fry it up to flavor some rice or what-have-you.
Best of luck!