I would like to use my shared web hosting on a big provider (like GoDaddy) as a proxy for web browsing. Is this possible? Any links or advice for how to get started doing this?
There's a PHP-based proxy named PHProxy / poxy at SourceForge.
I've successfully run this at GoDaddy, but you may run into bandwidth and script execution time-out limits for big files. Works OK for browsing sites.
If you're planning on using the proxy for anonymity/privacy purposes, I would suggest using TOR instead, since the GoDaddy logs may be seized and used as evidence in various jurisdictions.