File Manager (Want to change from ES File Explorer)
I've always like ES and Astro.
SMS/MMS Messaging (Using Messenger)
Google Messenger (from the Play Store) or Textra (ditto.)
Web Browser (Using Chrome & Dolphin)
I switched from Chrome on the desktop to Firefox, because Chrome just uses too much battery and RAM these days. So, I use Firefox, to sync bookmarks and even access tabs between devices. I'd use whichever browser (between Chrome and Firefox) you use on the desktop for that reason.
Twitter Client (Using Twitter & sometimes Plume)
I used to love Plume, but it's not so good now. For twitter I *really* want tweetmarker support, so that limits which clients I can use (and rules out the stock client.) Fenix is by far my favorite, Talon Pro is ok, Tweetings is also ok. All are better than Plume these days IMO.
Miscellaneous (Any other killer apps I need to know about)
Some people don't need this, but I use SMS Backuip+ (free) to back up my SMS/MMS messages to a label on my Gmail account, and also have it put all of my phone calls into a Google Calendar that I created. I love this app. Having SMS conversations in GMail makes them easier to search, and calls on my calendar is great so I can check when calls were made, etc.
Swiftkey for keyboards.
I love the Google Now launcher, though Action Launcher is a close second.
BeyondPod for podcasts (if you listen to those), though Pocket Casts is great, too.
Google InBox if you use Gmail - it makes processing email more efficient for me. If you don't use Gmail for your primary mail client, there are a number of good clients in the Play Store that are much better than the stock mail client. (K9, Aqua Mail, Type Mail.) And if you want something like Google's InBox for non-Gmail, Dropbox's Mailbox app is also great.
Flickr, Google Photos, or Dropbox for uploading photos to cloud storage automatically (so if you lose your phone, it breaks, etc., you won't lose any photos.)
Tapatalk (or, for here, the Android Central app) to access sites like this on the phone more efficiently.