ESP8266 Not Working Properly

pert:
As Juraj pointed out, you need to understand that when you have your ESP8266 connected to pins 0 and 1 of your Uno then you are using Serial to control the ESP8266 via AT commands. The trouble is that we usually use Serial also for sending debug output to the Serial Monitor. The ESP8266 may be confused by this debug output. The problem is made worse because there is debug output to Serial coded into the WiFiESP library, which is left on by default. I found that I needed to turn this off before the library would work with my ESP8266 connected to Serial. To do this you need to open WiFiESP/src/utility/debug.h and change line 32 from:

#define _ESPLOGLEVEL_ 3

to:

#define _ESPLOGLEVEL_ 0

It's unfortunate to lose all debug output when you're first trying to get the system working since you're basically working blind until you can get network communication going, after which you can use that for debug output if you like. I didn't have any problem with that but others are not so lucky with their ESP8266 ventures.

An alternative is to connect the ESP8266 to different pins on your Uno and use SoftwareSerial library. The downside of this is that software serial is not reliable at the default 115200 baud of the ESP8266 AT firmware. You can change the baud rate via an AT command, which would probably be easiest done by sending the command directly from the Serial Monitor, as you were doing before you started on your journey with the WiFiESP libirary.

It looks like you forgot the WiFi.init() call in your sketch. You should add the following line after the Serial.begin() call:

WiFi.init(&Serial);

I recommend starting with the example sketches you will find at File > Examples > WiFiESP until you have gotten everything working correctly to avoid problems like this. Note the example sketches are written assuming you will be using SoftwareSerial library on your Uno but they can be easily adapted if you prefer to stick with the connection on Serial.

Bleh, that was the problem, not calling

WiFi.init(&Serial);

I may look into SoftwareSerial (Which is now NewSoftwareSerial, right?)

But thanks so far!