How many years do I have to pay taxes on property before it becomes mine?

This is a misconception. The way to aquire property is to purchase it out-right from the owner. If the property has delinquent taxes owed, you may purchase the property for taxes at the annual Land Sale which is held the last Monday in August every year. If you purchase the property at the Land Sale, there is a waiting period of three years from the date of the delinquent taxes. If this period expires without the taxes being redeemed by the current owner, the property will mature to the purchaser. At this time the purchaser may request a tax deed on the property.
Example: If you purchase property for 2012 taxes  at the August 26, 2013 Land Sale, the property will mature to you on the last Monday in August 2015 if no one redeems the taxes before that time.