First thing I would do is disconnect the battery, and put your meter leads across the positive and negative wires going to the tractor and see what the resistance is. should be close to if not infinity. Something else you can try is fully charge the battery, but leave it disconnected overnight, and then before you try to start the mower, reconnect the battery and see if it will start. If it will start the mower then most likely you have a short. if it is dead without being hooked up, then most likely you have a bad battery.
After you get the cause determined, or problem fixed, you can disconnect the negative wire from the battery, and put the leads of your meter between the wire and the negative terminal of the battery, and the drain from the battery should be in the milliamp range, less than 100 milliamps.
Be aware that if you try the above procedure with a short in the system, the amp draw could very well exceed the capacity of the meter, and damage the meter or worse.
Post back with your finding, and somebody with good knowledge will be able to instruct you further.