What is the easiest way to upload video and audio content? I am creating a website archive that will allow users to upload various different types of media content including video, audio, images, documents and text. I want to make it easy for users to upload the media content to the site.

Is it better to upload directly to the website hosting itself or to use a third party video and audio hosting like YouTube and SoundCloud?

Either way how can it be made as easy as possible for the user to upload the media content?
Get a media management suite that integrates with as many third party hosting providers as possible. It means less work for you, less complexity, and less bandwidth on your server for which you have to pay.

Good suite's include, but are not limited to: 

Scald 


  Scald is a popular and flexible way to handle media in Drupal.
  
  Scald can handle images, video, audio, tweets and even Flash files.
  Scald works with YouTube, Vimeo, Soundcloud and other many other
  providers.


and Media


  The Media module provides an extensible framework for managing files
  and multimedia assets, regardless of whether they are hosted on your
  own site or a 3rd party site - it is commonly referred to as a 'file
  browser to the internet'.


For a front end media solution you can look at MediaFront


  The MediaFront module is a front end media solution for Drupal. It
  employs an innovative and intuitive administration interface that
  allows the website administrator to completely customize the front end
  media experience for their users without writing any code.