Ace of Arenas is available on the App Store for iOS devices, and on the Google Play Store for Android devices. To download the game, simply tap the "GET" button (for iOS) or the "INSTALL" button (for Android). Wait for the game to download, then tap the app icon on your home screen.
When you first start Ace of Arenas, you will be prompted to choose a user icon and enter a user name. You may also choose a flag if you wish. Once you've done that, the game will download some updates. You will then be asked if you would like to play the tutorial, and the game will download some more updates once the tutorial is complete (or immediately, if you choose to skip it). When that's all finished, you'll be free to battle to your heart's content!