and then make the wp-content directory writeable,
I'm not a GoDaddy user but if you can't run suPHP or change the ownership of those files and directory then you'll need to set wp-content to 777 .
(Drinks more coffee) This will require some explanation so I don't give out bad advice... anyone who sees a fault or gap (Wake up Otto and Whooami! :) ) please chime in.
I run my own virtual private server, so I make the files owned by myself and run the web server as a different userid (in my case www-data, it's an Ubuntu distribution). I'm the only one on my VPS with no other users defined or active.
My webserver can't write to the files because I set my directories to 755 which means owner (my userid) can read/write, others can only read so that rwxr-xr-x. The execute bit is required so that non-owners can enter that directory.
Files are set similarly as 644, which is read/write for the owner and read only for everyone else (rw-r--r--).
This is reasonably secure but has a problem: when I use the built in editor to upload images for my posts, or use the built in plugin upload management feature then I wont be able to upload files within WordPress. The webserver is running as www-data and can't write to the wp-content directory, it can only read the contents.
So to fix this, I make the directories that WordPress uses to be owned by my webserver. Since I administer that box, I can do that. If I did not admin that box, I would have to set the wp-content and it's sub folders to 777.
Using 777 is a compromise and it's not really safe. Any other users on your server could drop files/directories there; when you are the only one on a server or VPS it's less of a risk.
Try limiting 777 just to
wp-content/uploads
wp-content/cache
And see if you retain the ability to upload images and use them in posts. If that does not work then make wp-content and all of it's folders 777.
For enabling the plugin upgrade feature you will also need to make wp-admin and wp-include writable too (I have no idea why). I do not recommend you do this because your blog has a history of problems. So when a new version of your plugins are available, upload them the old fashioned way.