Fixing a flat on your lawn tractor is fairly easy. However, once a lawn tractor tire is compromised, it is likely to continue to go flat. The best permanent fix for any lawn tractor tire is simply to replace it. In the mean time, diagnose the problem and fix it temporarily so you can get out there and start mowing.
Remove the tire. First, use a car jack placed near the tire you wish to remove to lift the tire a few inches off of the ground. Next, detach the rim from the axle. For most lawn tractor models, this involves pulling off the hub cap by hand (or a pair of pliers if its stubborn) followed by a few washers. Refer to your lawn tractor's owner's manual for specific instructions for your make of lawn tractor.
Identify the source of the leak. Inflate the tire and then submerge it (while still attached to the axle) in water. Escaping air bubbles will point you to the leak.
Deflate the tire. Remove the valve cap and press the side of the valve with your finger until all of the air is released.
Remove the tire from the rim. First, you may need to break the "bead", or the rim of the tire which may be stuck to the rim itself. Wedge your screwdriver between the edge of the tire and the lip of the rim and press down to free the bead. Work around the circumference of the bead on both sides of the rim. Next, lay the rim flat on the ground to remove the tire. Then, pull the top bead over the rim of the tire. It may help to wedge the screwdriver under the bead to pop it over the rim section by section. Then slide the tire up the rim and pull the lower bead over the rim.
Repair a hole with your tire plug kit. Place two to three drops of rubber cement on the end of the probe. Stick it into the hole and then work the probe in and out to clean the hole and widen it to the size of the probe. Then add two to three more drops to the probe and apply it to the hole until it is coated with rubber cement. Thread the repair strip through the eye of the large needle. Then push it through the hole until only 1/4 inch of the tips of the repair strip are sticking out of the hole. Then trim the excess down to 1/8 inch.
Fix a leaking valve stem. The best way to do this is to buy a new valve stem to replace the old one. Remove the old valve stem by pushing it into the tire, then throw it away. Clean the hole and line it with a thin layer of grease. Then push the new valve through the hole from the inside of the tire out. Keep pushing until it pops into place and sits securely.
Buy a new tire. Any large tears in the tire can't be fixed. Neither can cracks or splits that are a result of dry rot in the tire. Tire sprays are unreliable and difficult to clean off of the tire.
Replace the tire on the rim. Lay the rim flat on the ground. Apply tire lube to both sides of the tire's bead. Then place the tire over the rim at a 45 degree angle. Pull the bottom of the bead over the lip of the rim starting with the lowest side. Then slide the tire down the rim and push the top bead under the lip of the rim.