Following is the list (regularly updated) of top-rated apps that are "ACTUALLY USEFUL" for most users and also aren't very resource hungry (Apps are recommended for devices with Android 4.0+).
- Launcher - Nova Launcher, (Prime)
- Browser - Naked Browser Pro, Opera Mobile (with Off-Road mode)
- Keyboard - SwiftKey, Flesky (No Swype)
- Alarm Clock - Alarmy (Sleep If U Can), Sleep Cycle alarm clock
- All-In-One (Multi-purpose) Apps - Smart Kit, All-In-One Toolbox
COMMUNICATION - - E-Mail - K-9 Mail, Mail.ru
- SMS - ZERO SMS, Textra, GO SMS Pro
- Instant Messaging - WhatsApp, Signal Private Messenger, BBM
- Video Calling - WhatsApp, JusTalk, Skype
SOCIAL MEDIA - - Facebook - Facebook Lite, Metal, Friendly or Mobile Website (with notifications turned on)
- Reddit - Reddit, reddit is fun, Relay for reddit (Pro), Sync for reddit (Pro)
- Forums - Tapatalk
PHOTO & VIDEO CAPTURE - - Camera - Open Camera, Cameringo+, Z Camera, Facetune
- Drawing - Adobe Capture CC
- Screen Recording - Screen Recorder, DU Recorder
- "Secret" Video Recorders - SVR, Secret Video Recorder
PHOTO & VIDEO EDITING - - Image Editor - Pixlr, PhotoDirector, PicsArt Photo Studio
- Photo Effects & Filters - PicSay Pro, SuperPhoto
- Stop motion Videography - PicPac
- Video Editors - VidTrim Pro
MUSIC PLAYBACK - - Audiophile Music Player - Neutron Music Player
- Simple Music Player - Pretty Good Music Player, Pi Music Player
- Stream Music (while screen is off) - Spotify, YouTube Music
VIDEO PLAYBACK - - Video Player - VLC, 321 Media Player, Video Player Perfect
- Youtube Video Player (Also can stream Music while screen is on) - Og Youtube, Newpipe
AUDIO APPS - - Emergency Alerts Radio - Scanner Radio Pro
- Melodies/Sounds - Relax Melodies
- Audio Enhancers - FauxSound Sound Control, ViPER4Android
- Music Recognition - SoundHound, TrackID™
WORK BASED - - Document-based Work - WPS Office, OfficeSuite, Polaris Office
- PDF Annotation & Markup - Squid, ezPDF
- Document Scanner - Office Lens, TurboScan, CamScanner, Business Card Reader Pro
- Book Readers - Moon+ Reader Pro, Aldiko Book Reader
- Jobs - Resume Builder Pro
NOTE TAKING - - Lectures - LectureNotes, LectureRecordings, Squid
- Calender + Notes - Calengoo, aCalendar+, ColorNote
- Save Notes For Reading Later - Pocket, Clipper Plus (with Sync)
INTERACTION ADDONS - - Notifications - Notific Pro, Touchless Notifications Pro
- Widgets - Beautiful Widgets Pro, Elixir 2 - Widgets, Multicon Widget
INTERFACE MODS - - Theming Engine - Subtratum, Arcus
- Interface Addons - Status Bar Mini PRO
- Display Mods for Sleeping - Twilight, Lux Auto Brightness
- LED Control - Light Flow Pro - LED Control, Light Manager - LED Settings
WEATHER - - Weather - Weather & Radar, 1Weather, Amber Weather
- Weather Reports (Detailed) - Weatherbomb
- Weather Daydream - YoWindow
FILE MANAGEMENT/EXCHANGE - - File Manager - Root Explorer, X-plore FIle Manager, ES File Explorer/Manager PRO
- Cloud Storage - Autosync Google Drive, Google Drive, Dropbox
- Data Sharing - Xender, SHAREit
PHOTO & VIDEO MANAGEMENT - - Gallery - QuickPic
- Photo Organizer - Slidebox
- Pinterest - Pinhog
REMOTE PHONE MANAGEMENT - - Manage Phone through PC - AirDroid
- Streaming (for DLNA/Chromecast, etc) - BubbleUPnP
- Remote PC Control - TeamViewer, Unified Remote, Splashtop Remote Desktop
GAMING - - Gaming - FPse, Sixaxis Controller
- Game Development - Coda Game
- Game Performance Enhancement - GameBooster 3, CM Gamebooster
SENSOR BASED - - Product Scanner - Google Goggles
- Maps - OsmAnd+, Google Maps, MAPS.ME, MapFactor, GPS Navigation & Maps Sygic.
- Sensors for Data Measurement - AndroSensor, SatStat, Sensorium, Ulysse Speedometer Pro, Runtastic Heart Rate PRO, GPS Data, GPS Data and Toolbox
- Cars - Torque Pro (OBD 2 & Car)
NETWORK - - Network Management - Fing
- Firewall - AFWall+, NoRoot Firewall
- Ad Blocker - AdAway, Tint Browser Adblock Add-on
SECURITY - - Call Recorders - Call Recorder - ACR, Call Recorder
- Caller Identification - Truecaller, whowho, Caller ID & Mobile Locator
- Anonymous Browsing - Orfox, Fire.onion, Orbot
- Anti-theft - Avast, Cerberus anti theft, Prey
GENERAL EDUCATIONAL - - Knowledge - WolframAlpha, All Wolfram Apps
- Maths - Math Helper, Formulae Helper
- Chemistry - Periodic Table
ADVANCED ENGINEERING - - Scientific Calculators - Mathematics, HiPER Calc, Graphing Calculator PRO
- Electronics - ElectroDroid Pro, Electrical calculations
- Unit Converters - Unit Converter Plus, Unit Converter, Unit Converter Pro Plus
- Astronomy - Mobile Observatory, Star Walk
- Patents - Relecura Patent Analytics
FINANCE - - Finance - XE Currency Pro, Barchart, INVESTING.com
- Personal Finance - Money Lover, Splitwise
BIOLOGY - - Medical - Handbook of diseases
- Food - E-Codes: Food Additives
LANGUAGES - - English - Vocabulary.com
- Learning New Languages - Duolingo, Memrise, Luvlingua Pro
PRODUCTIVITY/DEVELOPMENT - - Make your own Apps - Sketchware, Apper
- Tools - Smart Tools, Ruler+
- Programming - C4droid - C/C++ compiler & IDE
- GestureControl - GMD GestureControl
- Task Automation - Tasker
INCREASING BATTERY LIFE - - Battery Statistics - AccuBattery, BetterBatteryStats, GSam Battery Monitor
- AMOLED Display Mods for Battery - NOMone Resolution Changer, Pixoff: Battery Saver, Always On AMOLED
APP INSTALLATION/MODIFICATION - - Bloatware Removal - System app remover
- App Store - F-Droid, BlankStore, Aptoide Lite, AppBrain App Market, Fossdroid
- App modifications - Lucky Patcher (Remove Google License Verification and Ads from installed Apps)
- Install Custom Gapps - Open GApps
SERVICE/APPS MANAGEMENT - - Background Apps Management - Hibernation Manager, Greenify (Donation Package), ForceDoze, Naptime
- Task/App Managers - SystemPanel
- Apps & Services Management/Backup - System Cleanup, My Android Tools, Titanium Backup
- Root Permissions' Pre-requisites - SuperSU, BusyBox Pro
- Flash/Backup/Restore Kernels & Recoveries - Flashfire, Flashify
- All-in-One Kernel Management - Kernel Adiutor
- All-In-One ROM Customization/Backup - 3C Toolbox, ROM Toolbox Pro
Note -
Above mentioned are the Best Alternatives to Stock Apps for Android - "For High Performance and Efficient Resource Usage on Old/Legacy Phones", which have a Google Play Store rating of at least 3.5/5 (70%).