Without my chip inline, when I send something across the serial line, both the TX *AND* RX lines "pulse".

Is the VFD connected when this happens? It does send some stuff back to the host sometimes, maybe it's sending 'E' because you're writing stuff while it knows it's off.

I'm thinking that maybe I need to have the "/RES" line on the VFD connected to the +5Vdc. The sheet for the VFD specified that the VFD will reset when a low level signal is applied to the "/RES" line.

It does suggest (by the drawings) that it's supposed to be driven high, with no connection it may be floating and being interpreted as ground, so the display constantly resets. Or since it's not specified in the..generous..4 page spec sheet, it may not even turn on at all until it sees /RES high for the first time after power is applied.

Actually, does the VFD need to be reset everytime you turn it on?

Not besides /RES needing to be interpreted as high... it automatically initializes itself to a consistent basic state when it turns on.