Choosing Godaddy over Bluehost is a personal choice. I have experience with Godaddy and it's been pretty painless for me. In fact, I recently had an issue with Godaddy hosting, but it was resolved by the Godaddy CS rep who trolls Twitter and Wordress, answering support questions and getting things done. I am actually SUPREMELY IMPRESSED by Godaddy for implementing reputation management software and coming to the issues instead of responding to them. I'm so impressed, in fact, that I'm staying with Godaddy for some of my hosting accounts for that reason.
Now, let's get to the root of your issue. As songdogtech and others ascertained, when you moved to GoDaddy, you used their single click install to set up your new install of WordPress.
This puts your URL at http://www.mysite.com/wordpress/.
Godaddy usually puts a temporary HTML file called "default.html" in the top directory of your site, as a holding page until you replace that file with your own "index.html" or "index.php" file. They want something to be there so people that come to your domain don't see a blank page, right? So, you just delete or rename that file to something like "default.html+++" so that the server no longer recognizes it as a proper HTML file. This way, you can keep it there if you ever need it for any reason.
Now, in regards to changing your install so that you resolve at yourdomain.com instead of yourdomain.com/wordpress, that is not a complicated issue. You must follow the instructions Samboll posted here: http://codex.wordpress.org/Changing_The_Site_URL.
These instructions are new to the latest version of WordPress. In the past, we would move the index.php file from the /Wordpress directory one level up into the main directory, overwriting whatever temporary index file was there from a hosting company. Then, you would change the path in the wp-config file, as well as change the path in ADMIN-SETTINGS-GENERAL. But, it seems as if WordPress has created a new way to do this, so please follow those instructions.
What I'm wondering is if you exported your old database and want to import it into your new version of WordPress on Godaddy? You can export your old WordPress DB from Bluehost, then import it to your database on GoDaddy, so that you have all your posts and everything. You, of course, should move all your themes and plugins to the new wp-content folder as well, so everything connects up from the old database to the new one.
You want to read Moving WordPress to a New Server (http://codex.wordpress.org/Moving_WordPress) so you can understand how to get your old content into your new installation.
As far as databases go, when you set up the one click install, it should have configured a database for you. You must go into GoDaddy Admin panel, find your Database panel, then see if it set up a Database for you. If so, the Database has IT'S OWN USER NAME, PASSWORD and HOST NAME. These ARE NOT the same as your GoDaddy USER NAME and PASSWORD.
You must copy that information, and make sure that you add it correctly to your wp-config.php file. How do you access that file. You must login via FTP, using your URL, your GoDaddy account USER NAME, and your GoDaddy account PASSWORD. Download the wp-config.php file to a folder on your computer (remember where it is!) and then use a text editor or Dreamweaver to edit the file. When you open the file as a text document, you will see where you should be placing your database information.
You will also notice in that file a URL to retrieve authorization keys from WordPress that will help protect your site. You must copy and paste that URL in another browser window. The URL will generate a set of keys for you. You will copy and paste those keys exactly as they look in the wp-config.php file.
You also MUST REGISTER an ACCOUNT on WORPRESS.COM without fail and get your API key for AKISMET spam filter. If you haven't done so already, go to WordPress.com and just register for a PROFILE and not a BLOG. This will give you access to your special KEY, which you must enter into your WORDPRESS AKISMET plugin panel page. This will help you with filtering spam.
Hope this helps.