Each user will have a maximum of one image, which will probably be resized and saved as a thumbnail.
I was wondering where would be the best place to store those images, on the server or should I get some type of outside hosting. I have a shared hosting account (with hostmonster if that matters).
Also, what are some good image hosting websites?
Depends on how often it will be loaded. Once in a while is fine on your own server. Quite often should be on S3 or some service like that.
Completely depends on your traffic and the terms of your hosting contract. if you have unlimited bandwidth and storage why go anywhere else?
If it is just one thumbnail per user, your own hosting will likely be fine.
In the long run, you'll be considering new hosting for your website long before you'll consider outside hosting for images.