Pico / ESP8266 newbie (idiot?) question !

Your wiring sounds good - the red LED and WiFi network seem like a good start!

Actually, where did you get your ESP8266 from? They often have different versions of firmware on them, and different versions have different baud rates (the ones I sent with the KickStarter should use 9600 baud).

A simple way to check, which you can copy/paste onto the left-hand side of the IDE is:

Serial2.setup(9600, { rx: A3, tx : A2 });
Serial2.on('data',function(d) { console.log(JSON.stringify(d)); });
Serial2.write("AT+GMR\r\n");

That should output some text (you should also see the blue light on the ESP8266 flash). For example mine shows:

>Serial2.write("AT+GMR\r\n");
=undefined
"A"
"T+"
"GM"
"R\r"
"\r\n"
"00"
"18"
"00"
"09"
"02"
"-A"
"I"
"03"
"\r\n"
"\r\n"
"OK"
"\r\n"

Sounds like you tried with 115200, but maybe you could also try 57600 as well?

If that does work, you may find you still have some problems though. It's not just the baud rates that are different between versions :( However let us know how you get on - it might be there are some simple tweaks that could be made that would make it work.