Problem receiving full serial event+SMS
The arduino is a "small memory" device. So it probably does not have a big serial buffer. On say "linux on a PC" you get a 4096 byte buffer, but on an arduino, probably only say 16. Now as luck will have it "+CMGR: "REC READ" is precisely 16 characters.... 

From the code that I see, you seem to be sending commands to the modem and then reading/echoing them. I would have liked to use two different serial ports or something like that. 

In any case, if it sort of works for those first 16 characters, remove all the delays and put one of them near the end of the loop() function. Remove the initial print to the modem in the setup function.