How large of an area is your garden? If you planted in rows far enough apart to get a mower in, you might mow the paths so you can get to the plants and then start hoeing and pulling weeds from around your plants. If it isn't too big of a plot, this fall you could mow it all very close, cover it all with cardboard then mulch. I filled all my garden beds with mushroom compost and I would do the same in your situation. In my area (rural NE Ga mountains) it is available by the dumptruck load or, like I did 3 cubic feet in the back of my pickup truck. It is going to be a good bit of work whatever you do :(.

I am assuming that you are not using chemicals in your garden. I would consider that it is possible your neighbors are using them, which could be why they are weed free.

Dixie is right, if you have space, you need to work SLOWLY over the next few years to irradicate the grass and keep the clover in pathways that can be easily mown. It will be easier to get into and weed. Weeding is a must, it must be done frequently (once a week) - and keeping things spaced apart makes it easier to get in between.
Mulching will help immensely on all fronts: it makes weeds shallow rooted and easier to remove, and also smothers (using cardboard) beneath the mulch.

In a situation like yours, every time you till or turn over the soil you are going to uproot more seeds from the soil. It's best to mow, and remove anything you don't want and not to till.

http://www.cefs.ncsu.edu/resources/organicproductionguide/covercropsfinaljan2009.pdf

I have another great internet article I found on wide swath rows with cover cropping but I can't find it in my links right now. I'll check on it later and send it. Lots of good photos and it will be very helpful to you!

this might give you some ideas of how to manage things more effectively for your property...

all of the videos they have posted are really great for your situation. http://www.youtube.com/watch?feature=fvwp&v=7uBeUHAJYh8&NR=1

Thanks for the ideas so far. At first we were tilling, so from now on we'll start mowing. The garden is HUGE, I'd have to actually measure it, but it's the size of a large house. No, we aren't using chemicals in our garden. Call me naive, but it's a vegetable garden, so what sorts of chemicals can you use if you're going to eat what's planted in the soil? It's not like I can use Round-Up, because nothing will grow there then, right? I've never had anything but very small or container gardens, so I am really at a loss for weed control. I almost need a garden mentor to come in to help me set the thing up for next year, but I don't know where to look for one of those either. We live near Callaway Gardens, so we're close to the Alabama border. I thought maybe there'd be a gardening club in the area, but so far I haven't found one.

What about burning the weeds with something like a propane torch?

Controlled burns can work, however you've got trees close by....and they can also create more problems with other weeds. Some seed simply need heat and or smoke to start germination....

Chemicals are not an option for me, personally around food. Your choices are slim right now - basically hand weed around the plants that are growing, and then mow the areas that don't have food directly in them (yet) if you can, is probably going to be the easiest.
You will do fine, you will "get" this thing. Realize that it is simply going to take time, lots of time for you to manage this size of space. My thoughts now are to save what you can by doing what I mentioned above. In the fall, concentrate on a much smaller growing space. Lay cardboard down (it will take a LOT and if you can get free wood chips from tree trimmers, neighbors or anywhere, get them. Spread them on the cardboard and leave it all winter. Let it smother what is below.

But for now it seems your ambitions were larger than the amount of time you'd like to dedicate to working in a garden. That is the balance you are going to have to figure out...what you offer the garden in maintenance vs. what you have.

There are also spray options if you so choose to do that. You can mow an area now, spray it lightly with kill, leave it in the sun a few days-a week (most chemicals work with sunshine) and then go ahead and mulch REALLY heavily (especially with cardboard and wood chips). Next year plant sunflowers in that spot, the year after you can plant food. (sunflowers are used to clean up toxic areas).
You will however always have to watch the area - weed killer is only going kill the current weeds, not the seeds for the next year (or from past years). MULCH will be your answer, and it will make some lovely soil too :)

It's getting a little late to get much of a garden in. I would mow it and then till it about once every two weeks until most of those weeds died out. Put your cover crop in this
fall. Early next spiring, till a couple of times.

tilling will bring up more weed seeds. It's very harmful to soil structures.

Mow and till. It looks like you have some kind of large leafed squash growing in the front of the picture.