I suggest you use AltSoftwareSerial library - try the example program in post connecting-hcm375b-232-68n-with-arduino
initially run a loopback test - connect pins 8 and 9 characters entered on the serial monitor keyboard should echo back to the display
if that works then connect the ESP-01
remember the UNO uses 5V logic the ESP-01 3.3V logic - make sure you have a potential divider on the UNO serial Tx to ESP-01 Rx - see post arduino-mega-esp8266-esp-01s-sending-data-to-google-sheets for an example using a Mega
any particular reason to us a UNO? the IO on the ESP-01 is very limited but it can support simple projects, .g. reading a I2C interfaced sensor and uploading data to a website