Actually its very simple if your current host and godaddy hosting are using same control panel like cPanel, then cPanel has builtin feature to backup/restore accounts or to transfer the account, if the control panels are different still it can be done by manually taking backup of the files and also of the mysql databases and then restoring them with godaddy hosting, they might charge you some extra for the manual transfer, you can contact their support, as many hosting providers do that, charges extra for manual transfer and free with same control panel transfer.
As for domain, if your current registrar has put up no transfer lock on it, then will be simple you would need to make a transfer request with godaddy and put up the secret key (which you will get with your current domain registrar), once you have given that and paid for at least 1 year renewal, your domain will get transferred to godaddy registrar and also will get renewed for 1 year.
