I keep my veggies in those big "Lock & Lock" containers, the ones with the silicone seals and the flaps that keep the lids on tight. I've got 2 19qt ones, for parsley and cilantro, and a 23qt one for lettuce. They all have little trays at the bottom for drainage, and the 3 stacked together fit on the bottom shelf of my fridge.
The 19qt ones hold 3-5 bundles of parsley or cilantro, and the 23qt one holds 2 medium heads of leaf lettuce.
I rinse the veggies and put them in the containers, hold the lid on and shake a little bit. That seems to get a lot of the water off, and I can just pour it out. There's usually a little bit of water for a couple days, I just drain it off. The seals are airtight, so the veggies stay pretty fresh. (I can keep grapes fresh for 2 weeks in a similar container!)
If everything is *really* fresh when I prepare it, it will last about a week. (This is how long this many veggies last for me.) Most of the time I will find a few bad leaves here and there, when I am making salads in the morning, I will just toss those out as I go.
I let the containers empty out completely between uses, and rinse them really well before putting new veggies in. Every 3-4 weeks I will run them through the dishwasher to make sure they are sterile.
I used to just toss the veg in the fridge in the store plastic bags, but a lot of it went bad, it took up a lot of room, and it was a pain to wash it since I do salads in the morning. Now I just spend about 20 minutes rinsing everything, once a week, and just throw handfuls into the bowls. If I want to feed wet veggies, a quick swipe under the faucet does the trick.