i’m not going to write the entire procedure here since there are a lot of good articles out there.

here are some I always use when im migrating exchange servers from 2003 or 2007 to 2010 or 2013

so thank you Pete Long

For migrating exchange 2003 to 2013 you will need an additional step and migrate to exchange 2010 first.

The following links are very good to follow all migrations

Exchange 2003 to 2010

Exchange 2010 to 2013

enjoy and good luck