The ESP8266 is a 3.3V device. The UNO is a 5V device. The UNO voltages could damage the ESP8266.
Connecting UNO pin 0 (~4.8V TX) to the ESP8266 pin 20 (RX with ~3.6V limit) may have damaged the ESP8266.
There are lots of ways to do the level shifting. I prefer level-shifting modules, because they can handle higher speeds than some of the simpler ways (a few resistors).