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.
Removing sod doesn't work well if you have Bermuda grass. In this case, the way to get rid of the lawn is through solarization or herbicides, plus several months of waiting to be sure it's really gone.
Sheet mulching may also work, but you will have to do some hand pulling of any surviving Bermuda. Make sure not to plant until it is clear that all the Bermuda is dead. For fall and winter sheet mulching projects you will need to wait until late spring to be sure the Bermuda is all gone and not just dormant.
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.