If your lawn is really small, shovels and a mattock may do the trick. Otherwise, rent a sod-cutter to shave off the top grass evenly and a rototiller to create optimum soil.
Get rid of the old turf debris. Use shovels and garden rakes to remove what the sod-cutter can’t reach. Hopefully the old turf rolls up, but may have to come out in chunks. Use a wheelbarrow to haul it out of your way.
Do not simply till old turf in with the underlying soil. This can lead to a number of future lawn problems such as fungus growth and it makes it harder to grade the soil.
Level the exposed soil using rakes. If the soil is very uneven, consider backfilling low areas and leveling the area with a water drum roller. Once the area is level, add amendments in the form of compost. Cattle or dairy-based compost is preferred to deliver proper nutrients. Evenly spread about 1/4” thick over the graded soil. Avoid leaving big piles of compost. Use the rototiller to combine the soil and compost to get a healthy mix. Go to a depth of four to six inches.
Use the garden rake to grade the soil. Comb out any rocks, clods and debris. Try to grade soil about one inch lower than surrounding concrete driveways, sidewalks, and landscape edging so that the installed sod will be flush when installed.
Now you’re ready to lay sod. Rake out your own footprints as you go and avoid standing on fresh sod. Start with the border edges that run against concrete or landscaping. Make sure to tuck the edges tightly.
With the perimeter sod in place, determine the direction to roll the interior sod by identifying the general slope of the yard. Run the lengths of sod across the slope to minimize erosion and moisture loss. If possible, run sod along the longest dimension of the lawn, this makes for a faster install.
After lying the first interior roll, cut the second roll in half using a linoleum knife. Cutting on the dirt side of the roll will give you a straight, clean cut. Stagger the end seams to create a brick pattern. Offset end seams at least 18 inches for aesthetics. Lay one half and set aside the other for a later row. Continue laying the sod in this manner making sure to butt all seams tightly, almost folding them into each other. Pound the sod into place with your hand if necessary.