Apple Music was released as part of iOS 8.4 on June 30, 2015.
To check if your device is running iOS 8.4: Navigate to Settings -> General -> About -> Version.
To update to iOS 8.4: Navigate to Settings -> General -> Software Update.
Unlimited access to Apple Music is available for all iPad, iPhone and iPod touch owners running iOS 8.4 or later for $9.99 per month, or $14.99 per month for up to 6 people. The first 3 months of Apple Music is free. You can see a complete list of Apple Music features here.