I keep fridge veggies directly in the crisper drawer--no plastic bags. Moisture is the enemy.

Some things I wash the entire bunch/package and then don't use them all. Dry these as best as you can (either with a salad spinner* or let them sit out for a bit). Lettuce and berries go in tupperware with a couple of pieces of paper towel in the bottom to absorb any moisture. Herb bunches I lay out on paper towel in a single layer then roll.

*If you don't have one, my mother used to take the bunch of lettuce, wrap in a dishtowel and swing it around her head a few times.

Onions, potatoes, garlic, ginger are stored in a dark place with enough air circulation. I store easy to spoil onions like Vidalia the same way except I place individual onions in squares of newspapers, like an open bowl, so that their skin does not touch each other.

Leafy veggis are cleaned, pat dried and wrapped in paper towels - and then put in Debbi Myers green plastic bags.

Make sure your veggi bins in the refrigerator is clean and dry (I wipe with white vinegar). I also sprinkle baking soda on it and then line with paper towels before I put veggies in.

Potatoes, onions and winter squash - I store in a cupboard out of the light.

Lettuce - I wash, wrap in a paper towel, and store in one of those plastic clamshell containers (I save them when I but berries or prepped lettuce/salads).

Tomatoes - on the counter for a couple of days, otherwise in the fridge.

I store most other veggies in the fridge. I've also heard that cloth bags work well, but haven't made that leap yet.

My friend stores much of her produce in cloth bags and says the veggies stay a lot fresher that way.

I wish I had the magic answer to this, because I end up throwing away a lot of produce.

Onions, tomatoes, garlic are in a bowl on the counter. Bell peppers, squash, eggplant, etc can stay on the counter for a day or 2. Same with avocado, but I try to only buy ones that are ready to eat same day.
greens and cucumbers go in the fridge.

Do you do your shopping once a week, or every few days? I think if you can get into the mindset of only buying what you need over the next day or two, there should be room and/or you won't need the long term fridge storage.

I don't have enough room in my fridge for everything. Where do you store your vegetables? What will make them last longer?