I'm going to address the MediaWiki side of the question, and leave the "where can I find a good image hosting service" part for others:
There are several ways to allow the embedding of external images on your wiki. If you want to allow the embedding of external images from anywhere, just set $wgAllowExternalImages to a true value in your LocalSettings.php.
If you want to allow external images only from certain sites, you can either:
The latter method is somewhat more flexible, both because the whitelist entries are PCRE regexps, and also because the whitelist can be edited on-wiki by sysops.
All of these methods let you just write a plain URL like
http://www.example.com/image.jpg on a wiki page and have it be automatically converted to an image instead of an external link. (FWIW, only URLs ending in
.gif, in any letter case, are recognized as image links.)
As an alternative approach, you can also enable the $wgAllowImageTag option, which lets you include raw HTML
<img> tags on wiki pages. There are no whitelist restrictions on allowed image URLs in that case, though.
Also, you don't say what you find painful about MediaWiki image management, but I'm guessing it's the fact that the built-in upload interface only allows you to upload one file at a time. If so, note that there exist several ways to make the process more efficient, either by installing a MediaWiki extension like UploadWizard or MsUpload to provide an improved upload interface, or by using an external tool like Commonist to automate the uploading process.