I would start by programming the 08M to send a character out every second ...
#Picaxe 08M
#Terminal 2400
Do
SerOut 0, N2400, ( "U" )
Pause 1000
Loop
Run that and you should see "U" being printed ion the Terminal display.
Put that to one side and write a program for the 20X2 which reads serial and puts it to the Terminal display ...
#Picaxe 20X2
#Terminal 9600
Do
SerIn C.0, N2400, b0
SerTxd( b0 )
Loop
Leave the Programming Editor connected to the 20X2, Terminal open and power everything off. Connect your 08M leg 7 to leg 10 of the 20X2, connect the +V lines and 0V lines of both 08M and 20X2 circuits together, power up the 08M and 20X2 and you should should see the "U" displayed by the 20X2 program.