All videos has been working great for me for a few years! I play .mov files. But Chrome as you know stopped doing Quicktime support. And I'm moving up to Joomla 3 for mobile device support. So what to do:

Chrome: Won't play quicktime anymore

iOS: Won't play flash (flv).

A great solution would be realtime transcoding, but that's just my dream. What would you recommend I do? Convert my videos to what format? Or, is there a simpler, faster solution?

I would go for mp4 (h264 codec) videos.
As you can see they are the most supported format: caniuse.com/#search=mp4