I am just going through this exercise and I have had some troubles, but I have made progress.
To use softserial at say 19200bps, you must also set the baudrate in the ESP8266 to 19200. It will probabaly default to 115200.
I used a second USB-RS232 (TTL) module to momitor the soft serial ports & this was a problem.
Solution.
Make a direct connection to the USB-serial dongle to ESp8266 and use AT command to change the baud ratei
ie. AT+UART_DEF=19200,8,1,0,0
I could then see the communication between Arduino & 8266
AT
AT+RST
ATE0
AT+CWMODE=2
AT+CWDHCP=0,1
AT+CWSAP="RemoteXY","",10,0
AT+CIPMODE=0
AT+CIPMUX=1
AT+CIPSERVER=1,6377
AT+CIPSTO=7
And doing it again & monitoring the corresponding responses
OK
OK
OK
OK
OK
OK
OK
OK
I could then see the AP ok and I could connect to it. However, at the connection, it did not send any data and came up with a message
Error device does not reply using iPhone
Error Device not reply. Receive time out error. using Android
I did capture some data from the esp8266 back to arduino
+IPD,0,6:U��
+IPD,0,4:�
+IPD,0,6:U��
+IPD,0,6:U��0,CLOSED
Nothing is sent from the arduino to the 8266
SO that is where I am stuck at present.
lee