How much knowledge of the anime story for SAO do I need to know to get into the game since it seems to have the same protagonist?
Probably would help to watch first season of SAO anime for Hollow Fragment and second season for the Lost Song. Note that games have alternate universe stories so they are different then anime.
Hollow Fragment is a no brainer. It's over hundred hours of content, great combat and actually a decent translation in the PS4 version. Vita version is fine as well if you ignore the Engrish.
Lost Song I have on PS4 so no idea how is the Vita port.
Edit: Amalur is pretty awesome, at least on PC. The game has a ton of content, decent story and good combat.