The second chip receives all the commands from the first chip fine.
The serial output/input on a pic is TTL so you will have perfect communication in this type of setup. However a PC is RS-232 which is a 12V inverted communications signal. Normally you would use a MAX232 or MAX233 transceiver to convert the serial TTL I/O from the pic to RS-232. A lesser expensive option is to use a few transistors, a quick forum or web search should come up with a schematic.

However the "quick and dirty" way is by placing a 22K resistor in series with the TX pin on the PC side to the RX pin on the pic side. You need to also tie the PC serial ports GND to common on the pic side. (see "DEBUGIN" command in the PicBasic Pro manual for a schematic) BTW the Pic won't be able to transmit back to the PC with this method. To do that see the "DEBUG" command in the manual.

I have a 100K resistor going from the serial pin to ground (also tried without this).
I'm not sure that I follow what you mean here, see the above "quick and dirty" way.