Basically, transferring things from old_server to astahast shold be rather simple. I did not do it recently, but it should be rather straightforward.
1) On old_server, make a full database backup. Your server probably has it's tools for making the database backup, else you will have to learn how mysql does it.
2) Backup your whole site on your own pc, including the database backup. By the way, you should have done this every day, because in case of crash of your server, you need to have a backup.
3) Transfer all your files on Astahost.
4) Create your database on astahost. Remember your administrator names. I would suggest create on astahost new administrator names with the same names as on your old_server administrator names.
5) If necessary, check your php installation, probably some config files are false or need setup.
6) import your database, look your cpanel or read the mysql docs.
Hope this helped.
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