I just finished a Wordperss site that had a similar need for playing video. I would recommend going the Youtube route. Youtube will handle hosting, file conversions and with iframe embedding you can be sure the most devices possible will see your videos.
On the WordPress side of the equation, there are many helpful plugins that work with Youtube. My personal favorite is Tubepress.
Tubepress is great for making a playlist of just a few specific videos, or showing an entire archive of videos in pagination style. The customization available for displaying videos inline or in modal made it an easy choice for my custom theme.
If you’re quite sure you’re up to the challenge of hosting, properly encoding and serving video on your own, I’d recommend a two front assault. Use Flowplayer as your easy-to-use Flash player for older browsers, and then embrace HTML5 for the newer browsers.
The best possible resource for all things HTML5 Video is Dive into HTML5