Joseph, I am going to suggest 2 ways. Which one may be more useful, if at all, depends on what your soil is like and how quickly you want to get underway.
If you have a deep layer of topsoil (maybe 200-300mm or more), and the grass grows well, and you want to start immediately then dig spits of sod up and turn them upside down so the grass faces downward. You can break the sod up a little with a spade until it is loosish. The grass root will still be in the sod and this will take a little time to break down as the grass dies off. As it does the soil will loosen up. Much of the grass will die off as it is upside down, all of it won't. You will need to kill off the remainder the the grass either by chemical spray if you must (a glyphosate weed killer is safest although I personally wouldn't use a weed killer around my vege patch) or some form of mulch such as straw, hay, shredded leaves, mature compost, dried grass clippings that will block out the sun light and kill the residual grass. When you plant in to this, fairly rough, garden you would be best to use already started plants as seed germination may be haphazard and the grass may compete. When any crops are at a 1/2 decent height mulch around them. Veges planted from a tuber like potatos and garlic can go straight in. The mulch will break down and add organic matter top your soil which will also help break it up.
The other alternative mentioned is some form of raised or lazanga garden. You can either make this with ready to go ingrediants if you want to plant out immediately or you can slowly build it up and let things break down via nature for planting in your spring. This is good if your soil is shit.
If the former case you will need things already composted. Cardboard goes down first or you can use several sheets of newspaper, this kills off the grass. Mature animal poop or compost or leaves goes in. You can layer it down or simply scatter it well together. There is no one set way. Anyone who tells you it has to be a certain process is lying. Essential requirement however is that the organic material is composted or well rotted. If you can lay your hands on some free well rotted horse/sheep/cow/chicken poop or some nice mature backyard compost (check the maker has not added any chemicals or cat/dog poop in) and some nicely rotted leaves or some old pasture hay or straw or some matured mauchroom compost you are away laughing. Mix it all together and plant away. Just ensure the top 4-6 inchs is nice fine compost or top soil or something similar. You can plant (even seeds I reckon) straight in to that. It does not matter really how high you make the garden. I have put in a number of raised gardens using wooden sleepers 300mm to
400mm high.If you have time to let things rot down over autumn and winter then simply toss it all in and walk away for a season. Grass clippings, household food scraps, leaves, animal poops, hay or straw, partly composed compost, used coffee grounds, scrap fruit and veges from the local fruiter whatever whatever. If it is free and not tainted by chemicals take it. Some will tell you to use peat moss or lime, that costs money. You can likely all ingredients you need free as waste product. You not only divert waste from a landfill but you save your hard earned money for things like seedlings and beer. Coffee grounds from local cafes, fruit/vege scraps from fruit shops, grass either your own or a neighbours (just make sure they don't spray the shit out of their lawn. If so, check what with), leaves from around the neighbourhood, poop from stables or farms or hen houses, straw or hay from a local farm, 1/2 rotten compost from a neighbour who makes it but don't use it etc. It will break down over time mind and likely be 1/2 as high as the raw ingredients. DONT use bark chips. They take an age to break down.
Thus url shows the actual building process. You don't need to make your as high as this. They use lime and peat moss, I don't as it costs money. They layer it, I just bunged things in the garden and mixed it all up with a garden fork and left it over winter. I have planted in to my 'chuck and leave' raised gardens this spring and things are doing fine.
good luck, happy growing.
rob