In a nutshell, you want to follow this to install the esp hardware serial library, and tweaked header file, then use the following sketch that disables the debug stuff since you only have the one port on the Uno. SoftwareSerial doesn't work with the Cayenne serial handler unfortunately.
/*Cayenne ESP8266 Shield WiFi Example
Adapted from Blynk's ESP8266_Shield_HardSer Example
This sketch connects to the Cayenne server using an ESP8266 WiFi module as a shield connected
via a hardware serial to an Arduino.
You should install the ESP8266HardwareSerial.zip library via the Arduino IDE (Sketch->Include Library->Add .ZIP Library)
from the Cayenne extras/libraries folder (e.g. My Documents\Arduino\libraries\Cayenne\extras\libraries) to compile this example.
NOTE: Ensure a stable serial connection to ESP8266!
Firmware version 1.0.0 (AT v0.22) or later is needed.
You can change ESP baud rate. Connect to AT console and call:
For Cayenne Dashboard widgets using digital or analog pins this sketch will automatically
send data on those pins to the Cayenne server. If the widgets use Virtual Channels, data
should be sent to those channels using virtualWrites. Examples for sending and receiving
Virtual Channel data are under the Basics folder.
//#define CAYENNE_DEBUG // Uncomment to show debug messages
// Cayenne authentication token. This should be obtained from the Cayenne Dashboard.
char token = "";
char ssid = "";
char password = "";
//can't use debug as the Uno only has one serial port
//#define CAYENNE_PRINT Serial // Comment this out to disable prints and save space
// Set ESP8266 Serial object
#define EspSerial Serial
// Set ESP8266 baud rate
Cayenne.begin(token, wifi, ssid, password);
You wire the ESP up to the RX/TX lines as described in the sketch notes. GP0 and the ESP enables should be held high. Some require GP15 to be held low, etc. That's why I ask about which one you have.
If you try this, and it doesn't work, you probably have NodeLua on your ESP.