Take a web hosting plan with good space and bandwidth, checkout some good plans on www.canhost.us/hosting/shared-hosting.php However there are some free imagehosting sites like tinypics.com etc..
You should most likely get a small VPS, as many shared hosts will kick you out for hosting a hosting script.
they have limits, you'll have to contact them - but i wouldn't recommend godaddy hosting, and stay away from unlimited hosts, there is no such thing as unlimited, there is always a secret limit and when you pass that they will suspend your account you should go for a simple VPS depending on how many images you want to host, and that will give you more control and you will be less likely to be a victim of oversold packages
The cheapest way is to run it on free web host. The best way is to go with reliable hosting company. Read reviews in this forum and go with provider who offer everything what you need.