Having the same problem here, just got the game for PS4. Sitting there and holding R2 doesn't work, pressing it quickly doesn't work.
What eventually seemed to work was just mashing the R2 button for up to 30 seconds or so, eventually Geralt randomly fired off and Aard but that only got me to the last one and now he refuses to cast it...
Edit:
Oddly enough I finished the tutorial. When I set my unplugged wireless PS4 controller on my forearm while holding down R2 and pressing it with my left hand so that I could use my mouse to surf the web looking at sites for a solution. I noticed the sign would go off. When I put the controller back into both my hands and tried again, nothing. So there's something, for me at least going on with the wireless and its sending the R2 button commands to the game in certain positions. I"ll have to see if something could be helping interrupt it but all other inputs worked. All the dodging and parrying etc.... went through without a hitch, it's a very odd glitch.