This is a simple backup and restore process, with the additional step of making sure you backup the Reporting Services key, so you can restore that as well.
A number of links on the internet, but this one is a step by step guide with screenshots, much easier than trying to follow MSDN
or if you are only talking a small number of reports and you want to change the credentials,etc you could simply import the .rdl definition files from SourceControl (hopefully :) )
If the 2 domains can authenticate with one another, then the current credentials will work. otherwise you will need to change the account used.