The info on Vital Information is very good. I have a similar problem due mostly to my rooftop garden. I've pulled heads of broccoli, lettuce, zucchini, eggplant, and onions already this year and there's way too much to eat.
The lettuce will last a while if you can wash, dry, and bag in a zip lock bag, making sure you get as much air out as possible. My bags have lasted a week easily.
Broccoli will last 4-5 days if in bags in the fridge but a quick blanch (2 minutes), dry and freeze on a baking sheet before bagging in a freezer bag is very easy to do and will ensure that you'll have broccoli whenever you need it.
The zucchini and asparagus should last a few days. Zucchini is best in a plastic bag in the fridge and asparagus does well in the fridge, standing (with its root ends) in a little water. Both should last 2-4 days.
Your carrots, radishes and beets should last a while in the fridge, along with the cabbage but the spinach and peas will need to be eaten immediately.
Now to the cooking part. Knowing which vegetables to eat first, second, and which ones you can freeze, you should be able to put together some nice meals and eat well everyday. Fresh vegetables, to me, shouldn't be overcooked because they're so damn good. Your peas would do well as a side with a nice blanch and a little butter and salt. The pasta primavera idea is a good one - I used a zucchini, broccoli, and some basil from my garden, along with some left over cherry tomatoes and garlic to do this exact dish last night. I've been eating broccoli and zucchini pretty simply lately as well. The zucchini, I slice thin and fry with a little olive oil; when done, plate, salt/pepper, and parmesan and your good to go. Broccoli has been blanched, then tossed in hot olive oil with red pepper flakes, salt/pepper; or blanched with a little butter and salt; or blanched and topped with a yogurt, serrano, garlic, parsley mixture (one of my new favorites). Cabbage would make an excellent slaw or stir fry. With your radishes, you can eat raw with some salt (like my dad does) or you can sliced thin, salt and place on buttered bread or just add to salads (like the one you can build with your spinach).
I would reserve soups for the vegetables you're freezing or that have gone past the "fresh" stage. Just keep it simple and you'll enjoy your weekly bounty. Good luck.