Exporting Map With Individual Projected Shapefiles (?)

Hello,

I am trying to export a map from ArcGis containing individual projected files.

I have purchased shape files which I assigned a coordinate system in ArcMap. What I wish to do is define the projection and project all files at once, and export all ESRI shape files as individual layers to import into a different GIS application.

I seem to be having difficulty doing this. Might anybody know how I can accomplish this?

Thanks and hope to hear from you soon,
Katie