It's a glorious day to be a Nexus 9 owner, and not just because you probably love the tablet, but because you can now finally update that tablet to Android 5.1.1 through a couple of different methods. First and foremost, you can wait for the over-the-air update to hit your device if you don't want to mess around with flashing/installing the software manually to your device, and, you probably should just wait for the OTA if you aren't sure what the process of flashing involves and/or aren't comfortable with engaging in it. That being said, you can grab the OTA links straight from the Google servers or the factory images to get the latest software version of Android.
Now, there are two versions of the Nexus 9 tablet. The WiFi only model and the LTE model and the factory images, as well as the OTA download links are available for both of them, so no matter which version you have you can proceed with downloading one of them and getting it onto your device. This will bring the device up to build number LMY47X, and will bring about some much-needed maintenance, performance improvements and bug fixes. There are some visual changes, small ones mind you, as well as few new features. While Android 5.1.1 does include dual-sim support as well as HD Voice, those things won't be coming to the Nexus 9.
What will be showing up in this release are things like the animations for quick settings toggles, device protection, new animations in the clock app, volume bar and interruptions changes, changes to the screen pinning feature, as well as some small changes to other bits and pieces of the software like contacts having no colored overlay or Google+ links, and the ability to hide 'heads up' notifications without dismissing them entirely. If you're going to proceed with flashing the factory images or installing the OTA download links, you can follow this guide which should help you along with the process. You can grab both the OTA download links below, and the factory images for both models of the Nexus 9 are available here at the main page for all the Nexus device factory images.