I can't play Pokemon GO! What's wrong?

Most (if not all) of the problems listed in the question are related to the Pokemon GO servers being offline, slow to respond, or overloaded with requests.

In the meantime, nobody likes not playing the game. Fortunately, there are a few things you can do:

• Just Wait
  The easiest (and best) move to take is just to wait for the servers to restore to a normal state. You can check the server status at this handy page, or by just trying to log in to the game.
• Force Quit the App
  Sometimes, the app will get stuck in a cycle of failure and will be unable to actually do anything. You can fix this by [iOS] opening the Multitasking window and swiping up on Pokemon GO, or [Android] navigating to Settings > Apps > Pokemon GO and then touching the Force Quit button.
• [Android] Clear Caches/Data
  As a more extreme measure, you can go to Settings > Apps > Pokemon GO and tap the Clear Data button. Note that this will only delete your local data -- your Pokemon will be safe. However, you will need to log in again. Be sure you know what account you used/created for Pokemon GO if you do this.
• Reinstall the App
  If all else fails, you can just go the nuclear option and erase the entire app and start again. If you have an existing account (and you know the credentials used), your Pokemon will be at no risk and will all be there once you log in again.

Potential solution for gps, is be sure your location is enabled.

• On iOS, go to settings > Privacy > location services > Pokemon Go
  • Make sure it is on for "While Using"
• On Android go to settings > location and security
  • Make sure location is enabled

Another solution(s) that have helped individuals that you may want to try is:

• Turn on Wifi
• Reset Network Settings
• Turn phone on/off

Pokémon Go is frequently failing to login on my iPhone, same for my husband (Samsung) and even when we can login, on capturing or randomly, gym battles (1hp) etc, the game will just stop responding. We have WIFI call/WIFI in general also enabled on our phones. Now, we do have bad carrier signal at our home (but excellent WIFI), so easy to blame that, but... This is also the case should we pop into town where carrier signals are very strong.

However, (this may or may not get your Pokémon account banned), using Nox app player, I'm able to login no problem via my PC using the very same internet connection I use on my phone. I experience little/no connection drops and no client freezing, just annoying that I can't ( or rather, choose not to for cheating sake) move my location in game.

Not sure why, maybe someone can poke into this as it doesn't appear the servers are really all that 'slow to respond' and crippled on a frequent basis as is being advertised. It feels like the mobile platform in general has some kind of flaw preventing stable connections.