I carry a, iPhone 5s for work, so I have some insight...
If your Moto G seems really slow, there is something else going on... Did you get the 1GB or 2GB version?
TBH, for a low-mid range budget device, it is very quick and fluid, certainly as quick as my 5s. If you are getting errors and reload requests in the Play Store, that certainly seems like a connectivity issue... Are you using LTE or WiFi and have you done a speed test at all.
For apps, that is all over the board... You will find lots of recommendations out there, but it really depends on your usage. Some I recommend:
Google Photos - as an alternative to Gallery, it also can automatically backup photos to your Google account
n7player or vlc - for playing local music, Play Music is decent but these are much more feature rich for local playing of media files
Nova Launcher (Prime) - unless you really like the Google Now launcher, I recommend a 3rd party one, much more customization options (Apex, Next 3D, or any of a handful of others are excellent as well)
Snapseed - an old standby for Android photo editing
Evernote or Google Keep - for note taking and organization of those little bits of information (Keep is nice, ties to your Google acct and auto backup)
SwiftKey - A keyboard replacement that is highly customizable (Perfect Keyboard and Swype are nice alternatives to try)
Spotify or Pandora - Stream music online free
Facebook and Messenger - if FB is your thing
Google Drive - Online file storage, or your favorite one (DropBox, Box, etc)
Accuweather - Better weather and widgets than Google Now
GSAM Battery monitor - Nice battery monitor and widget for battery status
My Paid Apps - a nice tool to see apps you purchased from Google Play store, the native interface can make it tough to find these
Percent Enabler - Enable a little battery percentage number on the battery icon
RingDroid - Make/edit MP3 files to be ringtones
SMS Backup & Restore - Android does not have a native app to backup your text messages, this one will also push backups to your favorite cloud storage
SoundHound - identify music in seconds
Speedtest.net - Test internet connection speed
UB Reader - A nice eBook reader
ZIP Code Tools - Handy to lookup area codes, prefixes, zip codes, and other simple data
Of course, anything else you happen to be into... Starbucks, BestBuy, Newegg, Walmart, Flickr, Instagram, theScore, Google Translate, Allrecipes Dinner Spinner, Ingress, Pokemon Shuffle, blah, blah, blah... there are literally millions of apps, it is hard to recommend without knowing more about how you use your device and for what stuff.