Migrating from ArcGIS 9.x to ArcGIS 10 is straightforward. All your data (maps, layers, rasters, geodatabases, etc.) is directly readable in ArcGIS 10. However once you start using ArcGIS 10, you will need to save your documents (maps, layers, etc.) out in 9.x formats in order to share them with 9.x users. Use the Save a Copy command in ArcMap to create 9.x version of your map document. For more info on Save a Copy command.
Geodatabases can remain in 9.3.1 format and be used in 9.3.1 and 10. However, if you want to take advantage of the new geodatabase improvements in ArcGIS 10, you need to upgrade the geodatabase to 10, at which time 9.x clients will not be able to read it. In order to help with sharing data among various versions, the Create Geodatabase tool now allows you to create older versions of the Geodatabase so you can copy features from ArcGIS 10 into an older geodatabase to share with other users.