[AT&T Galaxy Note 3 Q&A, Help & Troubleshooting] DrAzzy:
I feel like for every-day typical phone use cases, there aren't that many gaps that stand out as needing to be filled by a third party app. You'll probably need a new music player, the defaults are crapola - but I can't be much help here; I keep my music as folders full of MP3 files (like God intended!) and play with Music Folder Player, and shun the dirty S&M ("sync and manage") apps.

There's a big list of stuff that I did when I first got the phone to get it to where I'm happy with it, but all of this was root required stuff. There are apps that come pre-loaded on android devices and can't be removed or disabled without root. Also, customizing stuff that can't be customized without root. I couldn't abide by some of the not-changable-without-root defaults. Getting rid of the "screen turns on so you can unlock it when you hit the home key" behavior is probably the most glaring example for me; my ass* was unlocking it constantly - but coming from iPhone, you probably view that as the Right Thing for home button to do.

I think most people use a custom keyboard. Swiftkey and Swype are the most popular. The phone comes with Swype, but it's an ancient version with limited support for changing the dictionary, you want the newest one.

I use Hangouts for gchat, but not for SMS. I actually prefer default app for SMS..

Settings:

Pull down status bar, long press on wifi, WiFi -> menu -> advanced settings, uncheck auto connect to AT&T hotspots (unless they don't suck where you are. Around here, they often don't have internet behind them, and just break your connectivity for the 30 seconds or so it takes for the phone to connect to them, determine that the wifi is no good, and fall back to 4G. At least this phone does that; Original note would just sit there like a fool trying to use non-functional wifi... I've also yet to find a hotspot that performs better than the 4G does. But I live in a city, so the situation may be different elsewhere).

Pull down status bar, long press on GPS. Uncheck "Use wireless networks" - on newer versions of android, this drains battery while idle; the damned thing wakes the phone periodically to check the networks around it - if the phone is nearly idle, this can halve battery life (!!!)

Pull down status bar, swipe to the left along the options at the top (woah, did you know you could do that?), tap multiwindow to enable multiwindow (this was a headline feature for the note 3, but comes disabled by default). If I long-press back button, that makes the multiwindow tab hide or unhide (once it's enabled), but I don't remember if I had to do something to get this behavior.

General must-have things:

AppOps Starter (to kill permissions from apps that request permissions you don't want to give them)

Battery Monitor Widget (gives you a realtime readout of battery usage)

ES File Explorer

If you want to pretty-ize clock + weather widgets:

DigiClock

Eye in Sky

If you work in IT or have need of these (if you aren't sure what these mean, you don't need them ;) ):

ConnectBot (telnet/ssh client)

AndFTP (ftp/sftp client, supports scp if you pay)

RDP Client (remote desktop and VNC client)

FEAT VPN (OpenVPN client)

Hacker's Keyboard (you'll tear your hair out trying to use swype/swiftkey for terminal sessions or RDP)

If you want to root the phone:

Root it, then immediately install titanium backup and "freeze" AT&T software update, so it won't update your phone (breaking root) while you're not looking

Titanium Backup - freeze all the useless bloatware that AT&T and Samsung included (there's a list of what apps you can freeze here: https://docs.google.com/spreadsheet/ccc?key=0AmzC8GFarItSdGpOZzBfWlR4MC03aDMyZ1BDUFNqU0E - not my work)

Root Explorer

Pen Window Manager for adding apps to the pen window feature

Xposed Framework

Wanam - general huge amount of customization, including selecting apps for multiwindow.

Keyboard Manager (almost a necessity if you use ConnectBot/RDP frequently - keyboards designed for messaging and such are garbage for doing administration of non-phone systems, while hacker's keyboard is garbage for messaging and everyday phone stuff)

Terminal Emulator (gives you a command prompt on phone. Doesn't require root, but there are very few use cases for terminal on android that don't also need root)

*Well, I keep it in my front pocket, so technically it wasn't my /ass/ unlocking the phone....