UPDATE: WORKAROUND AVAILABLE FOR THE “OUR SERVERS ARE EXPERIENCING ISSUES” MESSAGE
If you get the “Our servers are experiencing issues” message, you can *sometimes* get around this by closing Pokemon Go, disabling your GPS, then trying again. Once you’ve logged on, enable your GPS again to use the game!
Failing both of these I’ve found that sometimes doing a force stop, clearing game data and restarting can fix it also.
NOTE: That this doesn’t clear any of your progress as that is all saved on their server. Your settings will reset though, like sound might come back on. Also if you have limited data the game will then have to download all your game data again so just be aware of that.
Generally I’ve found if none of these steps work then the server is down.