You probably haven't heard much about eApps because they're not a huge player, but I haven't found anyone I like better. The customer service and support is top notch. I'm not sure about them actually assisting with the switch, but it should be pretty standard. You're going to FTP files over and relocate a database. If you're looking for help with this, feel free to PM me and I may be able to help you out on a consulting basis. Otherwise, post back here with questions and we have a huge community of experts eager to help you out.
Beware, all cloud hosts are not the same. That being said, the general idea is that you pool server resources to save money and allow for traffic spikes. Speaking very generally, cloud hosting is cheaper than VPS (virtual private server) hosting because you're sharing unused resources. As much of a fan of eApps and their VPS system, I'm not a diehard fan of their cloud offering. Maybe I'm old school but I like the level of control that a VPS gives me. Now, two explain the different types of cloud hosting...
Let's look at eApps cloud offering as it's the standard you'll find out there when you see the term cloud in the hosting industry. The cloud they give you is like an elastic server. If you're paying for 256 MB of RAM but suddenly need 512 MB, fine, they'll let you spike to 512 MB without crashing your site. That being said, it's a spike. Not a full time usage. This is a critical point.
Then there's a true cloud offering like Rackspace Cloud Sites. This is a cloud server where you start with a set number of resources but can load the server with a near infinite amount of traffic. Obviously it's not truly infinite but the theory is that they can handle anything you can throw at it. The offering from Rackspace is much more versatile and suited to high traffic websites. This is a good option to grow into as most websites won't need this type of resource usage out of the gate.
Does that help a bit? I raced through some of those definitions but I hope you get the basic points. If I were you, I would spin up a VPS with eApps (they start at $9/month) and practice moving your site over. You can test your site because you reassign your nameservers by using the static IP address assigned by eApps. Remember, that to signup for the VPS you need to go to http://cp.eapps.com.