First, you'll need to prepare your iOS device for the update, and the first step is backing things up in case you run into a problem. A full backup to iTunes is recommended, and it should be encrypted if you wish to preserve your Health and Activity data. Archiving the backup is also a good idea to make sure it remains available should you need to roll back from iOS 10.
Next, you'll need to install a profile on your device that will give you access to the beta software. It's accessible through Apple's Beta Software Program website, and once you've registered there you'll need to log in from the device you want to install the profile on.
If you've previously been enrolled in an iOS public beta or developer program, you may already have profiles installed on your device for testing beta software, and these should be removed through the Settings app before trying to install the new iOS 10 profile. It's possible you may also have a previous pending beta update such as a iOS 9.3.3 beta already downloaded to your device but not yet installed, and you'll want to remove that in Settings -> General -> Storage & iCloud Usage -> Manage Storage (Storage section) before proceeding.
Once the profile is installed and your device restarts, head to Software Update in Settings as with any other iOS update, and you should see iOS 10 Public Beta 1 available to install.
As you explore the public beta, you can use the included Feedback app to report bugs to Apple, and make sure to check out our iOS 10 forum to discuss your experiences with others and find answers to your questions.