I have written a few posts about videos and YouTube already. See them here: Screencast Tools, Video Creation PD, Embedding YouTube videos,
Our school has Google Apps for Education, so each student and teacher already has the ability to create a YouTube channel within his or her account. This is the easiest way for teachers and students to share videos. YouTube and all that comes with it is not always the best for everyone, though and YouTube is blocked at some schools.
Wistia is an easy video sharing site. The free account allows you 3 videos and 5 GB. This is not a lot but is an easy way for people to get started.
Edmodo is a free online management system that allows teachers and students to communicate, collaborate and share in a secure, Facebook-like environment. There is currently no limit as to the amount of videos you upload.
TeacherTube is a free education-based video site. There is currently no limit as to the amount of videos you upload.
All of these tools are easy to use. You might want to check with your school's technology director to see what is most feasible for your school's technology capabilities.