1. First, get a new account with us - and then transfer all of your website files.
2. Once that is done, you need to update your domain name's DNS/nameservers (with the company that registered your domain). That's all there is to it.
3. After your domain is completely pointing to your new hosting account, you could then safely cancel your old host.
We setup all accounts within minutes of receiving payment. Once you've updated your domain name's nameservers - it takes about 24-72 hours for your domain name to start working with your new hosting account. Do not cancel your old account until the new account has fully propagated.
If you would like us to handle the DNS changes for you, then transfer your domain to us before ordering your new account.