When I converted my personal sites, the main gotchas came from needing to use https along with http2. I had to go through all the templates etc finding http links and changing to https or no-protocol format links; etc.
The actual http2 bit was basically turn-key with my hosting company. From what I understand, browsers that don't understand http2 just get http1.
From memory I got ~20% speed boost for doing nothing more than enabling http2.