Is your site a complicated one or a simple one? If it's a relatively simple site, the easiest thing to do is to start by going to tools/export and export the database.

Then do a fresh install of Wordpress on your new host, install the theme, install any plug-ins you were using and then go to tools/import and import the content.

Job done.

If you have any custom widgets or have made changes to the theme, make a note of what they are (and copy the content of any widgets to a Word/notepad/OpenOffice document so you can copy it into the new site)

You may want to hire somebody on Fiverr to do the task for you, I had the same problem with moving my blog from one host to another, and rather me risking losing any files I just paid someone on fiverr to do it

I agree with Chris about Hostgator.

I recently wanted to move a Blog from one domain to another in Hostgator and their Support Operative was very helpful. If you want to preserve all your content, I would suggest that you download all your WP files and folders to your PC and then upload them to your new host, using ftp, eg Filezilla.

You will also need to export your associated SQL database to your PC and then add a new SQL database at your new host, making sure you give it the same name as it had at the old host. You do this in cPanel. There are some videos on YouTube that explain how to do it.

Make sure you back everything up and if you hit a problem ask Hostgator for help. They have a 24 hr Chat Support line.

You can do it yourself easily enough.
http://domainingdiva.com/transfer.pdf

You can use a plugin called WordPress Move to move everything from your old hosting account to your new hosting account. Or just submit a ticket and ask your new web host to do the transfer for your.

You might find this link useful:
Moving WordPress