[AT&T Galaxy Note 3 General] quicky2g:
I don't consider myself an Android expert. Just sharing my experience in hopes that it can help others.

Root info here.

After rooting, I installed Titanium Backup.

I referenced this post and this spreadsheet for possible apps to uninstall.

I'm sure there are others, but here's what I backed up and uninstalled with Titanium Backup:

ANT HAL Service 3.0.0
ANT Radio Service 04.01.00
ANT+ Plugins 1.8.0
AT&T DriveMode 1.0.2.5
AT&T FamilyMap 2.0
AT&T Hot Spots 0.974
AT&T Locker 1.1.12
AT&T Messages 1.0.0.11
AT&T Navigator 3.2.3.7230056
AT&T Ready2Go 1.5.2.2
AT&T Smart Wi-Fi 8
BlurbCheckout 1.0 (Build 125)
Browser Bar 1.0.3
Calendar widget 1.0
ChatON 3.1.25
Device Help 1.0
Games 1.30.0.30
Google Play Books 2.9.21
Google Play Magazines 2.0.0
Google Play Movies & TV 2.7.15
Help 1.0
HP Print Service Plugin 1.0.29
Internet 1.5
KLMS Agent 1.0
KNOX Store 1.0
My Files 1.0
My Magazine 1.0.0
My Magazine Here and now 1.1.0
PEN.UP 0.9.5
Picasa Uploader 1.3.242047
PickupTutorial 4.3
S Health 2.5.0.58
Samsung account 1.4.0321
Samsung Apps
Samsung Apps Widget 2.02.21
Samsung Calendar SyncAdapter 1.2
Samsung Cloud Data Relay 3.03.003
Samsung Cloud Quota 1.3.1
Samsung Contact SyncAdapter 1.2
Samsung Contents Agent 1.0
Samsung Hub
Sumsung Link 1.9.0901
Samsung SBrowser SyncAdapter 1.2
Samsung SetupWizard 1.0
Samsung Video
SamsungHub Updater
Scrapbook 1.0
SIM Toolkit 4.3
SNS 1.2.0
Story Album 1.0.02
Story Album Widget 1.0
Travel wallpaper 1.0
TripAdvisor 0.0.8
Video Editor Lit 1.042
Video Player 1.1
Weather Widget
WlanTest 1.0.0
Yahoo! Finance
Bubbles 1.0
Camera Test 1.0
com.sec.enterprise.knox.attestation 1.0
com.sec.knox.eventsmanager 1.0
POLARIS Office 5

I tried uninstalling a few apps that caused problems:

Calendar
Calendar Storage
Samsung Push Service

I don't like the default calendar app layout. I use aCalendar instead but it seems to need the default calendar apps. I restored the default calendar and storage apps with Titanium Backup and everything started working fine again.

I use File Manager (Explorer) instead of the MyFiles app, MX Player instead of the default Samsung video player, 1Weather instead of the default weather widget, and Kingsoft Office instead of POLARIS.

I'm still trying to find the package for the default picture viewer app so I can uninstall it. I use QuickPic instead.

I keep an eye on my CPU and memory stats with Elixir 2 and the widgets.

I got rid of startup apps with Startup Manager.

Recently I started using Greenify to keep apps from running in the background.

I use CPU Tuner to improve my battery life.

I installed AdAway from F-Droid to get rid of ads.

The performance and battery life are better on my phone than the day I got it. The CPU usage used to idle at over 30% when I watched it with Elixir 2 widgets. Now it idles at 5% - 15%. I gained a huge amount of RAM from uninstalling so many apps and using Startup Manager and Greenify. I used to hover around 300MB of free RAM, but now I have 700MB - 1.1GB most of the time. Every once in a while I run the 1 Tap Boost from Clean Master to get some of my RAM back, but not nearly as much as I needed to when I first got the phone.

Looking forward to hearing about other apps people uninstalled without causing problems.