I buy all my veggies, one week's worth at a time. I rinse everything, and store it in large tupperware-like containers. (They're made by Lock&Lock.) My containers have a raised grate-sort of thing, so water can drain a bit and the veggies won't sit in the water. I can keep head lettuce, parsley, cilantro, kale, and dandelion greens fresh in these containers for over a week.
As Rue mentioned, your refrigerator temp can make a huge difference. My old fridge had some issues with maintaining a constant temp. I got a new fridge about a month ago, and it's fabulous! My produce and other foods stay fresh MUCH longer now. (I found some parsley in the drawer from 2 weeks ago, and it still looked fine. I tossed it into the compost, but it was still firm and bright green.) If you don't have one, a refrigerator/freezer thermometer is fairly inepensive, and I think it's worth it to check the actual temperatures in your fridge.