1. Play Store
2.
Lookout Security, and Samsung makes a program for backup:
http://www.samsung.com/us/kies/
3. Just run the Dropbox app, it should walk you through the rest
4. I'm only two months into using Android as my main phone compared to some people who've been using it for years (I've played around with it on tablet before and my brother's phone, though), so I may be missing a few well known ones, but here are the apps that are "essential" for me right now:
Waze - great and free GPS app with live traffic updates (done by the community who uses it)
QR Droid - for scanning QR codes
JoeVLC - a very good video player, though some people prefer MX Player
Snapdragon BatteryGuru - optimizes battery life automatically after learning your habits after a few days (works on all Qualcomm Snapdragon powered phones - mine is an HTC One, but it will work on the non-Exynos S4's, as well)
Adobe Reader - for PDFs
Pandora/Google Play Music/Spotify - I'm currently on Play Music with "All Access" subscription, and use free/ad-supported Pandora
SkyDrive/Google Drive/Spideroak - for cloud storage, though you can use Dropbox or Box
Amazon Appstore - they have a free app a day, mostly games
Netflix - if you have a subscription
Nook/Kindle - I have both since I have ebooks on both
Authenticator - for 2 factor authentication with email or other online logins, really essential for security, currently have it enable for Gmail and Outlook
Advanced Task Killer - allows you to temporarily close apps that might be hogging memory
File Manager - basically a file explorer, very useful for managing and moving files on your phone
Alarm Clock Xtreme Free - can have multiple alarms, and I use it so I can be awoken to mp3 files I have on the phone
SwiftKey/Google Keyboard/Swype - Google Keyboard is free, the other two are paid; I use SwiftKey because it's got good correction/prediction, but I also like the look of the electric blue lettering on black keys in the "neon" theme (some people like Flow/Swype/gesture based typing, but personally I preferring tapping)
Poweramp - my preferred music player, since it has a lot of features and an equalizer (free 2 week trial, but for free there is also Winamp)
Wifi Analyzer - good graphical way to see the strength of WiFi networks around you, also lets you see their strength over time
Speed Test - SpeedTest.net app, a good way to routinely test how fast your connection is over WiFi or cellular data
Google Earth/Google Sky Map - mostly because they are pretty, but also useful and educational
Firefox - I don't use it all the time on my phone because the stock browser on the One is actually faster for me and uses less memory, but it might be useful for you (especially with addons/plugins, just like the desktop version of Firefox)
There are some apps that require root (like Greenify) or some custom ROM apps/enhancements, but I personally haven't rooted or flashed my phone since I'm waiting to see what happens with the "Google Experience", Nexus, or whatever they call it Google version of the One (you might be in same boat with the S4) before undertaking any modification to stock.
5. Google Now is Google's predictive "cards" engine. Whenever you search by text or voice (you get to Google Now by pressing and holding the home button) Google analyzes it. Certain behaviors you do over time will trigger Now to automatically give you notifications and "cards" which have that information laid out. For instance, it will give you the weather via notification, or if you search for an address in Google it will give you directions, or tell you what the traffic is on your route to/from work.
It can also do things like show you flight times the day of your flight if you happen to have a confirmation email or itinerary in your email inbox. Google Now basically tries to give you the information you want as you need it or right before you need it. If you are familiar with Apple's Siri, it's that but with certain types of automatic notifications.
P.S. I highly recommend listening to the
All About Android podcast. They air live in audio and video every Tuesday.
Not only do they discuss Android news, but every episode they recommend 3 or 4 apps. It's how I got a little up to speed as to what's going on in Android, since I was a Windows Phone user until recently.