Last spring we did the corrugated cardboard box, pizza box and newspaper thing over the grass and lasagna style topping of topsoil, straw and composted cow manure, then added mulch over that. We created a raised bed of about 8-10 inches. During the growing season we fertilized with fish emulsion and also added kelp and bloodmeal and some lime. We had tomato plants over 7 foot tall and they produced like crazy. Two things that didn't work well were peppers and zucchini. Powdery mildew on the squash, most likely from over watering. The green beans did very well.
The cardboard started to breakdown about the end of summer, and the soil was much improved below the beds. This year we are hoping for better results. In the fall we added alfalfa pellets, bone meal and coffee grounds topped with leaf mold and some compost. I have
compost in a tumble that has been working since the fall and should be ready in time for the spring planting.
Overall, the cardboard worked well in keeping the grass and weeds to almost nothing. The mulch used on top was Sweet Peet. It is a recycled and composted horse bedding that works really well and breaks down within a year.
I hope you have as good a garden as we've had. We had not had a garden in about 15 years and this worked out so well we are devoting more space this year. Can't eat a lawn.