The Google Stadia controller will not be able to seamlessly connect to all your devices at launch. While you will be able to utilize a wireless connection with Chromecast Ultra, you'll have to plug the controller in with a USB-C cable to use it with a computer or Google Pixel 3 phone. You'll still be able to switch devices on the fly, however.
While the trailer says the Stadia controller connects directly to the Stadia servers, the fine print in the trailer spotted by 9to5google clarifies that this will only be possible with Chromecast Ultra at launch. The controller is the only piece of first-party hardware made directly for Stadia and will feature Stadia-specific shortcuts. Stadia supports third-party controllers and mouse and keyboard too.
The Google Stadia controller is also planned to have Bluetooth support down the road, but this too won't be available at launch. It does include a 3.5mm headphone jack for wired use though, as confirmed on a Reddit AMA. You'll still be able to play with Bluetooth headphones if you are running Stadia on a computer or a Pixel 3 phone by pairing the headphones with the device you are using, but Chromecast Ultra users will have to wait until controller support is added.
Stadia's launch will also have limited mobile support, only running on variants of the Google Pixel 3. Support for other mobile phones is planned but not yet available. Theoretically, you might be able to run Stadia on your phone's Google Chrome app, but this isn't confirmed to work.
For now, Google appears to be focusing its efforts on the big-screen experience. While it may be disappointing that Stadia is launching without a few of it's advertised features, Stadia isn't even in full-launch yet. Only the $130 "Founders Edition" will be available on the November 19th launch date, which includes a controller, Chromecast Ultra, and two three-month subscriptions to Stadia Pro (one to give away to a friend) along with Destiny 2: The Collection. After that, Stadia Pro will cost $10 a month.
Stadia Pro is the premium version of Stadia, which supports 4k resolution and 5.1 surround sound. According to the Stadia trailer, Pro users will also have access to certain games for free. Stadia Pro is the only service being offered at launch, with the base version launching for free in 2020.
Stadia will be launching in 14 different territories, including the U.S. (minus Hawaii and Guam), U.K., and Canada with 31 announced games.
Sources: 9to5google, Reddit