Setting Up an Image Hosting is easy.
You have to know the basic of webhosting and DNS, if you fail to do this, I recommend you to hire a freelancer.
1. Buy A web hosting
>Most of shared hosting provider doesn't allow you to use their hosting server for a file/image storage.
I recommending you to buy a small VPS server
2. I will not explain how to setup the servers, vps, etc. You can search google or this forums. You will found so many result.
4. You can point your name server in your co.cc domain names or create new A records to your destination server.
5. Image Hosting Script
This is also free, you found it on google or a script directory.
4. Upload and install the script, Configure permission (chmod)
PS. I prefer to host my file/image in S3 services than using a free image hosting like tinypic, photobucket.
Any questions ?