From my experience with the Amiga serial port...
You get 19200 Baud on an Amiga (A500/A1000/A2000) with 68000 CPU (@ 7MHz) using standard serial device (WB 1.2 or WB 1.3) and decent terminal program (eg. JR-Comm), 38400 Baud can be achieved with faster serial device and tweaking screen mode (number of colours), etc.
A plain A1200 gives you 57600 Baud using standard serial device (WB 3.1) and decent terminal program (eg. NComm), with any accelerator (020/030 CPU) over ~30Mhz you can do 115200 Baud.
The "hardware" part of RTS/CTS refers to the physical
RS-232 pins in the serial port to handle the handshake, as opposed to XON/XOFF handshaking which is "soft" as in characters sent by software in the data flow.
I haven't done any serial programming on the Amiga, so can't help much there.