The fastest way to remove a lawn is to physically remove the sod by cutting it into strips with a sod cutter. Sod cutters can be rented, or you can hire someone to do the work. The sod can then be rolled up and taken away. Removing the sod generally removes the top layer of soil, which is the most nutrient-rich soil in the yard, so you will need to build the nutrients back into the soil with compost.
If you have Bermuda grass, either method will require additional work. Bermuda grass sends out runners above and below the soil surface. If any part of these runners is left behind, they can regrow. These need to be pulled or covered with more cardboard as soon as they appear. If you take care of Bermuda grass regrowth as soon as it reappears you can exhaust its reserves and it will not comeback. If you let it regrow too long, it will continue to be a problem in your yard.
An alternative to removing the strips of sod from the site is to use the strips in the new garden and “compost” in place:
You can lay them upside down in new landscape area to create a mound, as fill in a low spot, or to create a berm.