UPDATE , Feb. 28, 2021: definitely check out Aaron Blaise's YouTube Channel for the many additional animation tutorials he has posted in the past six years since I first made this post. You can learn a lot about animating and using the TVPaint interface just from watching his Livestreams of him animating scenes using TVPaint.
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Here's a free sample of one of Aaron's animation tutorials using TVPaint:
https://www.youtube.com/watch?v=i0i36AlJciI
Also see Aaron's recent short film "Trunk Troubles" with Cyrano the Elephant , animated in TVPaint:
https://www.youtube.com/watch?v=6i9jJsPH-L0
Also, be sure to subscribe to Aaron's YouTube Channel . He has other free animation tutorials posted there:
https://www.youtube.com/user/AaronBlaiseArt/search?query=animation+tutorial+
Including this great overview of the basic functions for animating in TVPaint: