You can do this with an EventTrigger.
You can define the trigger in a FrameworkElement.Triggers property of any container of both the button and the animation targets.
<StackPanel
Orientation="Horizontal">
<StackPanel.Triggers>
<EventTrigger
SourceName="TheButton"
RoutedEvent="Button.Click">
<BeginStoryboard>
<Storyboard>
<ColorAnimation
Storyboard.TargetName="LimeRect"
Storyboard.TargetProperty="Fill.Color"
To="Red" />
<ColorAnimation
Storyboard.TargetName="RedRect"
Storyboard.TargetProperty="Fill.Color"
To="Lime" />
</Storyboard>
</BeginStoryboard>
</EventTrigger>
</StackPanel.Triggers>
<Button
x:Name="TheButton"
Content="Play" />
<Rectangle
x:Name="LimeRect"
Fill="Lime"
Width="50"
Height="50" />
<Rectangle
x:Name="RedRect"
Fill="Red"
Width="50"
Height="50" />
</StackPanel>
If there is a relative path to your targets, you can use Storyboard.Target="{Binding PathToTarget}"
in place of Storyboard.TargetName="TargetName"
.
EDIT: (see comments)
If you are animating the button itself, you can put the triggers right in the button and you don't need any target names.
Example - Animating the size of a ToggleButton:
<ToggleButton
Content="Toggle"
Width="50"
Height="50">
<ToggleButton.Triggers>
<EventTrigger
RoutedEvent="ToggleButton.Checked">
<BeginStoryboard>
<Storyboard>
<DoubleAnimation
Duration="00:00:00.2"
Storyboard.TargetProperty="Width"
To="100" />
<DoubleAnimation
Duration="00:00:00.2"
Storyboard.TargetProperty="Height"
To="100" />
</Storyboard>
</BeginStoryboard>
</EventTrigger>
<EventTrigger
RoutedEvent="ToggleButton.Unchecked">
<BeginStoryboard>
<Storyboard>
<DoubleAnimation
Duration="00:00:00.2"
Storyboard.TargetProperty="Width"
To="50" />
<DoubleAnimation
Duration="00:00:00.2"
Storyboard.TargetProperty="Height"
To="50" />
</Storyboard>
</BeginStoryboard>
</EventTrigger>
</ToggleButton.Triggers>
</ToggleButton>