What's the best way to weed my scruffy yard?

I have a smallish back yard with a slight slope. When we moved in a couple of summers ago it was overgrown with weeds and totally unkempt, I am afraid to say it remains that way. Every spring I have great intentions about cleaning it up and planting a garden and grass. Every spring I get totally overwhelmed with the amount of work to be done.

What is the best/most efficient way to get rid of all these weeds? It is a decent sized area and I would like to plant some veggies so I would like to avoid an herbicide.