How do I include a delay only for the first tooltip and not for subsequent tooltips in my app?

The trick is to use the InitialShowDelay and BetweenShowDelay property appropriately as shown below in all your tooltips:

[XAML]

<Ellipse Height='25' Width='50'
Fill='Gray'
HorizontalAlignment='Left'
ToolTipService.InitialShowDelay='1000'
ToolTipService.ShowDuration='7000'
ToolTipService.BetweenShowDelay='2000'>
<Ellipse.ToolTip>
<ToolTip Placement='Right'
PlacementRectangle='50,0,0,0'
HorizontalOffset='10'
VerticalOffset='20'
HasDropShadow='false'
Opened='whenToolTipOpens'
Closed='whenToolTipCloses'
>
<BulletDecorator>
<BulletDecorator.Bullet>
<Ellipse Height='10' Width='20' Fill='Blue'/>
</BulletDecorator.Bullet>
<TextBlock>Uses the ToolTip Class