When moving from one Hosting Environment to another, the best course of action will always be to make sure that you're transferring to is either identical to the one you're leaving. I know that's of very little help right this second, but it's something to keep in mind for the future. That said, let's move on to the problems!
5xx Errors
These can pop up as a result of a number of things, though when it comes to WordPress & Shared Hosting, they generally pop up as a result of 1). An issue with your .htaccess file or 2). Your WordPress installation is hitting the oh-so-dreaded "resource-wall" that is common with shared hosting.
A "resource-wall" would be that which the web hosting provider has set to limit the amount of resources an account may consume in a period of time. This includes CPU & CPU Cycles, RAM/Memory, Input/Output (IO) Operations, INODES (number of files and/or directories an account may create/host), Bandwidth in Mbs or Kbs (Megabits per Second / Kilobits per Second). When this wall is hit, your account is forced to slow down by the server software. This either results in slow service or, more often than not, 503 errors (i.e. Service Unavailable).
Redirection Errors
Redirection Errors can generally be narrowed down to two things, 1). The .htaccess file or 2). A Plugin/Theme isn't playing nicely with GoDaddy's hosting environment (and most likely isn't going to).
How Can I Fix & Prevent This From Happening?
You'll need to find your root directory. Most likely this is either ./public, ./htdocs or ./public_html (most common). This is the directory where your main index.php file will be and it's also where you'll find your .htaccess file.
- Download a Backup Copy of your .htaccess.
- Delete the .htaccess file inside of your root directory.
- Head to yoursite.ext/wp-admin/ and try to log-in.
If you're successful, look to the left sidebar menu and find Permalinks. All you need to do is click the link and then click submit (found towards the bottom of the page); no need to change any of the settings. Why? Just by clicking on submit, WordPress will automatically regenerate a cleaned up .htaccess file for you and place it in your web root. This ensures that you now have the most up to date & accurate .htaccess file for your installation.
If the errors persist beyond this point, look to the web root once more and see if an "error.log" file has been generated (or a file with a similar name). If so, follow up with the last 10-15 lines starting from the bottom of the file and moving towards the top. That'll often provide specifics as to what file, or what files are causing issues. We can then either disable the plugin(s) associated to see if that fixes the issue, or take another course of action.
WordPress Plugins & Disabling Them All
Typing, when there's an issue, it's best to simply disable all plugins and enable them one-by-one until you find the one that is causing issues (more so for when you can't find the exact cause from looking through the error log). This can be time-consuming, though it'll provide definite information.
To do this, you simply need to move everything in the ./wp-content/plugins/ folder elsewhere. You don't have to delete the files, simply move them using your S/FTP program, or using the File Manager that should be available through GoDaddy's control panel (which is cPanel IIRC). Once you've moved the files, refresh any page and that should deactivate them all. You won't lose any data as a result as this doesn't delete theme, nor the data associated with their installation, it simply marks them as disabled, meaning they won't run when you access your site.
As a result, depending on what you have installed, things may look a little odd and other errors may pop up (especially if you're using a theme that relies on functionality from one of the plugins), but seeing errors as a result of need instead of incompatibility or worse, something unknown, is a good thing. These can be resolved by simply moving the plugin folder for that plugin back to ./wp-content/plugins/. Ultimately, you'll refresh, move, refresh, move, rinse and repeat until things stop working.
When they stop, you stop moving folders. Identify the last plugin you moved and it should be safe to mark that one as an issue. Move the problem plugin back out, refresh to make sure that fixes the issue and then continue with the rest until you hit another problem.
Hopefully you either won't have to do that too much, or better, hopefully we'll be able to find the issue with the error log or the issue is resolved once a new .htaccess file is generated :).