First, a disclaimer: As with any beta software, you may run into some issues and odd behaviors with MacOS Sierra that should have you thinking twice about jumping in now instead of waiting for the final release to arrive this fall. If you can't wait, I suggest you install the beta on a secondary Mac if you have one or, failing that, on a separate partition on your primary Mac.
I don't have an extra Mac lying around, so I created a partition for MacOS Sierra on my MacBook Pro. It's easy to accomplish in a few easy steps:
1. Launch Disk Utility
2. Select the disk you'd like to partition from from the left panel and then click the Partition button.
3. Click the "+" button below the Partition Layout diagram.
4. Choose a size for the new partition by dragging to resize a wedge of the pie chart that represents your hard drive (MacOS Sierra is only 4.86GB) and give it a name. You can leave the Format as OS X Extended (Journaled).
5. Click Apply and wait a minute or two for the new partition to be created.
With your backup Mac or partition ready, head to the Apple Beta Software Program site. You will need to sign up to take part in the program, if you haven't already. The process takes only a minute or two and requires your Apple ID.
Now, here's the part where we remind you again about the potential dangers of using beta software and the importance of backing up your Mac before installing Sierra (Apple will do the same this when you arrive at its Beta Software Program page). With Time Machine, Apple couldn't make it any easier to create a backup. Go ahead, we'll wait right here.
With your Mac freshly backed up, click the blue Redeem Code button on the Apple Beta Software Program site, which launches the Mac App Store. You'll need to use Safari in order to open the Mac App Store automatically. With Chrome or another browser, you will need to open the Mac App Store and redeem the code manually.
You can check on the progress of the download from the Purchased tab in the Mac App Store.
After Sierra downloads, you will need to choose where to install it. If you created a partition, click Show All Disks from the installer window and select your partition before clicking Install. (Then to choose which partition to load, hold down the Option key when you Mac boots up.)
Remember, the purpose of a beta is to iron out any kinks in the software. If and when you encounter an issue, report it to Apple using the Feedback Assistant, which you'll find via its purple icon in the Dock.