Sounds like you're talking about a Flash Animation. I haven't created one in a while but I remember it was a steep learning curve using Flash. An easier to use application at the time was called Swish click here but I see it has moved on considerably since I last played around with it, although it does have an excellent online community backing it up, so you always have plenty of help. What you describe is pretty basic stuff in Swish or Flash, so give it a go.

A less 'professional' approach might be to use an animated gif of your logo.