How do I transfer to a new host?

I’m trying to transfer the website for the Kentucky Coalition to Abolish the Death Penalty from Earthlink.net to BlueHost.com. I’ve already bought the server space on the latter, but need to transfer over my existing site. Just exporting it through Tools > Export > All content won’t work as I’ve made a lot of customizations to the template.

How do I transfer over the entire website? Earthlink.net doesn’t support this install of WordPress. BlueHost said I need to download the database, but I’m not sure where I can find it.

Thanks for any help you can provide!