Things I install when I get a new Android phone:

First, install Nova Launcher to replace the default launcher.

Install SwiftKey keyboard.   (alternatives)

games tab

• Minesweeper
• Carcassonne

main tab

• communications
  • Google Hangouts
  • Google Voice
  • Facebook Messenger
  • Twitter
• GMail *
• Evernote
• Screen Filter

location-based tab

• Google Map *
• Weather Channel

time-based tab

• Calendar *
• Clock *

system utilties tab

• Tasker
• Terminal IDE
• External Keyboard Helper Pro
• Ultimate Rotation Control
• FX or ES File Explorer
• Zedge

root-only apps tab

• ROM Toolbox Pro
• DiskUsage
• Game Killer

ringtones

• notification — Zedge > search for {lowport quiet}

filetype mappings / file viewers

• PDF viewer ⇒ Polaris Office
• videos of many different codecs ⇒ MX Player

* built-in app