Hosting externally.

I’m a part of my school’s 2013/14 Journalism class. Our current setup is horrible, and the teacher (though retiring this year) is allowing me to completely overhaul the site as the next Online Editor In Chief. I’m already experienced in WordPress as a regular blog member, but not an admin (however I am in the process of learning it).

However, my teacher is only allowing me to host it through Bluehost because we already have a setup with them and security and all that. I haven’t yet pitched to her the wordpress hosting, but she has already expressed that she vastly prefers we continue our final year with Bluehost and then it can be changed to whatever after my peers and I graduate next year.

Basically, how would i go about hosting externally right off of the bat? we already have a domain name that won’t be changed (unless my teacher miraculously agrees to use WordPress hosting). I noticed that upon attempting to initially set up a wordpress blog it gave me the options of free (.wordpress.com), and premium and business.