I can’t add any transitions to my timeline in Sony Vegas Movie Studio Platinum 9?

Try Vegas Pro 9. Movie Studio 9 is really basic and I wouldn’t rely on it.. I Hope this helps
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Well, you have to actually merge 2 clips together. If you don’t do that then the transition has no where to go.
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First of all, disregard the advice to go to Vegas Pro to solve this problem. Vegas Movie Studio does about 90% of the things Pro does. Pro adds unlimited tracks and the high-end features needed in broadcast and movie production applications. Vegas Movie Studio can do almost anything a home or semi-pro user needs. Transitions are a very basic part of movie editing and all versions of Vegas can do transitions.

In order to have a transition you first have to have a crossfade. A crossfade is created by dragging the two events so they overlap. The amount of overlap determines how long the crossfade will last. Try dragging the beginning of the second clip into the end of the first clip and you will see a balloon appear telling you how many frames of overlap you have. Once you have set the crossfade, you drag a transition onto the overlapped area. You may actually have an overlap, but simply are not dragging your transition onto the overlapped area. That will cause the red circle to appear. Zoom into the overlapped area to see it more clearly and drag the transition over the area where you see the crossfade. Then the transition properties box will appear and allow you to adjust the properties. If you want to accept the default properties, just close the properties dialog box.

As a final note, make sure you are dragging the transition over a video track and not an audio track.

You can get a lot of free Vegas training online. For example, go to YouTube and search for "Vegas Tutorial" and the feature you are looking for.
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I train students in Sony Vegas. See an example of my Vegas editing at http://www.youtube.com/watch?v=zV3RK61igvo