Vimeo is the most reliable hosting site for video. They have very clear online tutorials, free and pro levels of membership, and great embedding.
Of course as the founder I'm biased, but you should use our WordPress hosting company: http://WPEngine.com.
As for dealing with large files and delivering quickly, just upload them to S3. If you need a CDN (which you probably don't), something like MaxCDN can be configured in front of S3 for faster, world-wide delivery for a decent price.