I have my own idea to transfer…
Purchase the account on Hostgator.com.
Add domain and install wordpress on it. Ask support for the ip login of your wordpress you have installed. after getting that login to that wordpress.
Now Go to your website:
Go to Tools->Export all the files from the wordpress which depends on the size of your website. It may be of 10-20MB in size. Before all this install all the plugins to the new server.
Now import that file to the new server via import option. Select all files and media there. Now install theme and customise it.
The files you have imported download all the files from the old server to the new server without any interruptions.
Now all you have to do is to point you domain to the sever. You are on to the new server without downtime.