The most economical thing is to gather up your fallen leaves in the fall with a mulching mower and till them in.

That's a good sized garden! The above is a great idea for next fall, but for this year you might check your dump and see if they have compost (I'm lucky to live in an area where the dump compost is certified organic) or a landscape materials place (those are also organic in this area, but more pricey)
You might want to peruse the soil and compost forum for lots of ideas for next year that will be cheaper than buying. Composting, sheet or lasagne composting, cover crops and much much more! Nancy

This is the way I do all my beds:

-mow the grass as low to the soil as possible

-smother the grass with leaves,newspaper,cardboard whatever is adequate to smother and kill the underlying grass

- cover the newspaper/leaves/cardboard with compost/topsoil.. Basically making a raised bed

- mulch heavily with a couple inches of leaves,etc

-plant!

That simple.. You could be planting in a couple of days.. The only expense is the compost/topsoil you would fill the beds with.. But if you have a compost pile, everything is free.. Also lookout for your city giving away free compost.. That were I got mine from.. I would of preferred my own compost, but inhad such a large area, the compos pile had to be is big as my yard! I did this with over 1,260 square feet of beds.. Vegetables,strawberries,blueberries, flowers... It works like a charm!

Best of luck,
Joe

Joe, as you have made it clear in other threads that you never disturb the ground in any way with any kind of tool, could you give a hint as to how to plant! into sod? Because it sounds like your advice here is to cover sod with cardboard and etc and then before anything has decomposed begin planting.

Also, perhaps you missed that this OP mentions that his is a second-year garden so presumably it is not presently established grass sod. Also it is unclear how cardboard will on a short time-scale amend the soil beneath. Within days or weeks we expect nutrients from compost etc to leach through the cardboard and have an effect on the ground beneath it?

Ohio is not northern VA. There is a climate difference at least. In the latter place it's past time to plant spring crops, so the OP needs to amend the ground ASAP if he wants to have a chance to grow peas and lettuce and other cool-weather crops. Covering with cardboard won't do that, nor will it warm the soil. If he has not added lime in the past then the ground is surely acidic, and is the likely cause of last year's poor performance. The most effective tack to plant now is to make the rows or planting holes directly into the clay, incorporate goodly amounts of wood ash and compost, and plant!