How to connect an Arduino to WIFI using an ESP8266

Sending the following commands to the ESP should get you started:

AT+RST                         # Reset
AT+CWMODE=1                    # Set to station-mode as a client
ATE0                           # turn off command echo
AT+CWLAP                       # optional: list access points
AT+CWJAP_DEF="SSID","PASSWORD" # Always connect to this SSID

(I made-up the comment syntax; don't try to send those.) The full ESP8266 AT Instruction Set has the complete set of commands and a functional description of each one.

Update:

... do I need to connect RX of ESP8266 to RX of Arduino, TX of ESP8266 to TX of Arduino, Ground of Arduino to RESET ?

Rx & Tx must be cross-connected with the corresponding Arduino pins and the ESP's reset pulled high (3.3v).