Update the installation documentation page

docs/getting-started/installation.md

@@ -1,9 +1,10 @@
 # Installing uv
 
-Install uv with our standalone installers or your package manager of choice (e.g.,
-`pip install uv`).
+## Installation methods
 
-## Standalone installer
+Install uv with our standalone installers or your package manager of choice.
+
+### Standalone installer
 
 uv provides a standalone installer to download and install uv:
 
@@ -41,18 +42,7 @@ $ curl -LsSf https://astral.sh/uv/0.3.1/install.sh | sh
 $ powershell -c "irm https://astral.sh/uv/0.3.1/install.ps1 | iex"
 ```
 
-!!! tip
-
-    When uv is installed via the standalone installer, self-updates are enabled:
-
-    ```console
-    $ uv self update
-    ```
-
-    When another installation method is used, self-updates are disabled. Use the package manager's
-    upgrade method instead.
-
-## PyPI
+### PyPI
 
 For convenience, uv is published to [PyPI](https://pypi.org/project/uv/).
 
@@ -75,29 +65,84 @@ $ pip install uv
     [contributing setup guide](https://github.com/astral-sh/uv/blob/main/CONTRIBUTING.md#setup)
     for details on building uv from source.
 
-## Homebrew
+### Homebrew
 
 uv is available in the core Homebrew packages.
 
 ```console
 $ brew install uv
 ```
 
-## Docker
+### Docker
 
 uv provides a Docker image at
 [`ghcr.io/astral-sh/uv`](https://github.com/astral-sh/uv/pkgs/container/uv).
 
 See our guide on [using uv in Docker](../guides/integration/docker.md) for more details.
 
-## GitHub Releases
+### GitHub Releases
 
 uv release artifacts can be downloaded directly from
 [GitHub Releases](https://github.com/astral-sh/uv/releases).
 
 Each release page includes binaries for all supported platforms as well as instructions for using
 the standalone installer via `github.com` instead of `astral.sh`.
 
+## Upgrading uv
+
+When uv is installed via the standalone installer, it can update itself on-demand:
+
+```console
+$ uv self update
+```
+
+When another installation method is used, self-updates are disabled. Use the package manager's
+upgrade method instead. For example, with `pip`:
+
+```console
+$ pip install --upgrade uv
+```
+
+## Shell autocompletion
+
+To enable shell autocompletion for uv commands, run one of the following:
+
+=== Linux and macOS
+
+    ```bash
+    # Determine your shell (use `echo $SHELL` to find the shell you are using.):
+    echo 'eval "$(uv generate-shell-completion bash)"' >> ~/.bashrc
+    echo 'eval "$(uv generate-shell-completion zsh)"' >> ~/.zshrc
+    echo 'uv generate-shell-completion fish | source' >> ~/.config/fish/config.fish
+    echo 'eval (uv generate-shell-completion elvish | slurp)' >> ~/.elvish/rc.elv
+    ```
+
+=== Windows
+
+    ```powershell
+    Add-Content -Path $PROFILE -Value '(& uv generate-shell-completion powershell) | Out-String | Invoke-Expression
+    ```
+
+Then restart the shell or source the shell config file.
+
+## Uninstallation
+
+If you need to remove uv from your system, just remove the `uv` and `uvx` binaries:
+
+```console
+$ rm ~/.cargo/bin/uv ~/.cargo/bin/uvx
+```
+
+!!! tip
+
+    You may want to remove data that uv has stored before removing the binaries:
+
+    ```console
+    $ uv cache clean
+    $ rm -r "$(uv python dir)"
+    $ rm -r "$(uv tool dir)"
+    ```
+
 ## Next steps
 
 See the [first steps](./first-steps.md) or jump straight to the [guides](../guides/index.md) to