Firstly, one dosen't simply walk into mordor, I mean copy databases. You need to do this right. You'll want to make sure there's no ongoing writes, then make sure everything is copied in a ACID manner. You could shut down the database, move it, then get hit with the realisation things went horribly wrong cause they're different versions...
Luckily mysql comes with a tool to do this called mysqldump - and there's a question on SO on using it with lots of syntax examples - It should be something like
mysql -u<user> -p < db_backup.dump. I would also recommend checking the versions since you may need to run an upgrade script between versions. Move the DUMP file, don't copy the database. Run the dump, run the upgrade script, test. As you can tell, Mysql has good resources on it, so read up and make sure you know what you're doing.
If you're using postgres, there's also pg_dump (which has an oh so elegant syntax). And its file level backup documentation is true anywhere - since shutting down a server is the surest way to not have writes while you're backing up, mucking up consistancy.
To move the dump file while there's quite a few methods SFTP is probably the ideal combination of speed and security. You can transfer files computer to computer with pscp or download them to your windows system, then reupload them with cyberduck. Since you already have SSH, this should be the simplest way to do this.
Web server configuration, and files (IE, the effective contents of /etc/ and /var/www) should be alright to copy, since they arn't constantly being changed.