Hi bknights,
Many shared hosting do not allow image hosting or image sharing scripts. You will need to check that if you want to host images on shared hosting.
For a simple website you can choose a VPS hosting to host your images. But you will need to manage all image urls to come from VPS. In addition, you will have to manage a VPS for hosting images.
One more way is to host your images in cdn. In this way also you will need to change all urls of images to cdn urls. You can use cloudflare, which gives free and premium access to cdn. This way your website will load much faster with images on cloudflare cdn.
For a wordpress site, you can use a free cdn for hosting your images. If you are laready using Jetpack, a Wordpress plugin, it have a module photon, which makes your site faster by hosting your site images on wordpress cdn servers.
Cloudflare can be used as a cdn for any website, simple or wordpress. It caches your website and keeps it on many servers located in different locations. When someone opens your website in browser it will get the pages from the nearest location and this will make your website pages much faster.
Thank you