Two years ago we were in a similar situation. We had just moved to a new house, and the entire backyard was essentially a lawn. What we did was quite simple - remove the grass and a good chunk of the topsoil, fill it up with horse and chicken manure (which you can get pretty much anywhere), till it, and voila - your soil is ready. The past two years we had an absolutely wonderful garden with lots of yield.

If I remember correctly, we would do this in early April, right before we'd start putting in the first seedlings. When you google manure, you get a lot of questions about the safety of manure (because of E.coli etc.), but the answer is typically "don't worry". The USDA apparently recommends an "application window of 120 days prior to harvesting and eating any vegetable where the edible part comes in contact with the ground". So doing this in early spring makes a lot of sense.

Btw, I'm in zone 6 too.