The targets for effects in a text object in InDesign are: the object as a whole; the text frame's fill (if any) and stroke (if any); and the text itself, which includes any applied stroke. In the current object model, the fill and stroke of the text are not separated to allow different effects to be applied to them.
If you convert the text to outlines, you can then apply separate fill and stroke effects because they are now native vector drawing objects, like any other graphics frame in InDesign.
If you need to keep the text editable, copy the text frame and paste in place so you have duplicate text objects one above the other. Put them on separate layers if you are in a version earlier than CS5, to make it easier to select them. Give the text in the lower frame the color you want, but no stroke. Give the upper layer the stroke, but no fill color.
In Photoshop, you can apply a stroke to text as an effect, but you can't then add additional effects to the stroke (such as bevel and emboss).