There is no advantage to having never applied before. It is a standard lottery system and the only "advantage" is a bonus point for every year you get drawn. This bonus point however, is almost nothing and barely, if even at all, increases your chances.
In order to hunt, you gotta go to the website
www.azgfd.com/ and apply.
A real quick lesson is this. In AZ, you can hunt most small game with a simple hunting license, available at any WalMart or most gun stores. There are times when the season is closed for a specific species however, so it would be best to consult the website or pick up a copy of the rules at the nearest Wallyworld.
To hunt big game (Elk, deer, javelina?, Antelope) you must apply and be drawn. The Elk, Antelope and Deer draws are soon, and you should not wait, if you really want to try hunting big game this year. A javelina draw is later in the year, as the hunts for them are scehuled in March.