The client can be installed and updated via the Homebrew package manager for macOS and Linux. You will need to install brew and then you can install talosctl from the Sidero Labs tap. 
brew install siderolabs/tap/talosctl
This will also keep your version of talosctl up to date with new releases. This homebrew tap also has formulae for omnictl if you need to install that package.
Note: Your talosctl version should match the version of Talos Linux you are running on a host. To install a specific version of talosctl with brew you can follow this github issue.

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Alternative install

You can automatically install the correct version of talosctl for your operating system and architecture with an installer script. This script won’t keep your version updated with releases and you will need to re-run the script to download a new version. 
curl -sL https://talos.dev/install | sh
This script will work on macOS, Linux, and WSL on Windows. It supports amd64 and arm64 architecture.

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Manual and Windows install

All versions can be manually downloaded from the talos releases page including Linux, macOS, and Windows. You will need to add the binary to a folder part of your executable $PATH to use it without providing the full path to the executable. Updating the binary will be a manual process.