I am no expert on this stuff yet.
I was running my WP site on a VPS with W3 Total Cache plugin using APCu. To be honest I wasn’t getting anything flash.
After a little research and a post on the forum similar to yours Best WordPress setup? FastCGI, HHVM (PHP5 Backup), Pagespeed? I deciuded to go with REDIS. I haven’t compared it to FastCGI yet as I don’t have enough time to change to compare the very small difference.
Too my knowledge, choosing REDIS, it will do page and object caching. Browser (static) caching is done in NGINX I use EE it’s awesome if you’re not already using it.
Database caching isn’t done but from my research actually slows down TTFB etc so really isn’t worth it. Also I found with W3TC object caching slowed down the backend dramatically. REDIS does seem to slow down the backend slightly but nothing like W3TC, I can live with it.
So to answer your question assuming NGINX does static caching, REDI does page & object. FastCGI only does page caching I believe, so you would need something else for object caching.
On http://www.webpagetest.org/ I am now getting .8s TTFB from Australia (server in Singapore) and First Load Time of ~3s @ 1.6MB & 50 requests the repeat load time down to ~2s.
I am very happy with this.
I also have a heap of plugins and WooCommerce running sure to slow things down.