Answer: Please keep in mind that most garden areas will require regular maintenance, more so while becoming established, so this might be a consideration in how much space you switch over.
You could possibly smother the existing grass beneath a layer of newspaper topped with organic mulch. This method retains any existing topsoil and increases the organic matter in the soil when the paper and mulch break down over time. Use five or six sheets of damp newspaper topped with a thick layer of mulch. Overlap the edges of the paper to exclude light. If any perennial weeds come up through, spot treat with an herbicide containing glyphosate. Once the lawn is smothered, you can loosen the soil with a tiller, or by spade, or dig individual planting holes depending on what you are planting.
Another option would be to use the herbicide over the entire area, then loosen the soil and add organic matter, then plant. This would be faster than smothering but if the soil is thin and rocky you may have trouble tilling it up. (Stones jam the tiller.) If you do this be sure to read and follow all of the label directions on the herbicide including waiting period.
I would not recommmend using plastic or landscaping cloth. The plastic excludes both rain and air and intensifies the sun so it is harmful to the soil. The landscape cloth is difficult to plant around and prevents mulch from breaking down naturally to feed the soil on an ongoing basis as it should. Debris becomes caught on top of the fabric and then weeds will root down into it and become entangled in the fabric. If you are mulching correctly, fluffing and renewing it as needed, there is no need for a fabric.
Q&A Library Searching Tips