ESP8266 communications stopped working with Arduino Uno

Hello all!

A couple weeks ago, I was able to program an ESP8266 (ESP-01) using my Arduino Uno. It stopped working, and after hours of trying different things, I bought an ESP8266 programming USB. It works wonderfully, I can program from the Arduino IDE and monitor the serial from it as well. When I take my now known to be working ESP8266 and plug into my Arduino, I get no comms back - I am certain it is wired properly.

But I'd still like to get it working with my Arduino. I suspect that something must have happened with my Uno. I did put new sketches on my Uno, and I'm thinking the Sketch might be not pulling in streams from the ESP8266, but it's just a theory. Is there a way to poll from the ESP8266 serial and print it through my Uno?

Uno:

void setup() {
  // put your setup code here, to run once:
  Serial.begin(9600);
}

void loop() {
  // put your main code here, to run repeatedly:

}

ESP8266:

#include <ESP8266WiFi.h>
#include <WiFiClient.h> 

const char* ssid = "NAME";
const char* password = "PASS";


void setup()
{
  Serial.begin(115200);
  Serial.println("Starting");
  WiFi.begin(ssid, password);
  while (WiFi.status() != WL_CONNECTED)
  {
    delay(1000);
    Serial.println("Connecting to WiFi...");
  }
  Serial.println("Connected to WiFi!");
}

void loop() {

}

remove Serial.begin from the Uno sketch

Please explain what the function of the Uno will be. Just a programmer/TTL-to-USB converter? Or is it supposed to process data from the ESP8266 and next show that on the PC? Or ...

Please provide a schematic/wiring diagram how everything is connected; photo of paper/pencil drawing is fine (and usually preferred over Fritzing).