Ok, finally had some time to sit down and check this out again.
It was definitely good to measure each item separately and check with a voltmeter.
I was probably wiring a couple of things wrong, and luckily it didn't seem to hurt anything.
The sensor was outputting -4mv with a null signal. Then the signal turns '+' for most of the output range. The datasheet mentions -40mv at resting, but in other places I've seen -4mv.
I only then connected up the INA122 and got up to nearly the 3.3V output. I was worried about the -4mv being amplified, but it seems to start from zero and go '+' from there. So, it looks like this combo works pretty well. Signal seems stable with me just blowing pressure into the tube.
Now, I'll pneumatically connect up to a blood pressure cuff and I'll post a pic of the pressure signal when I do. Cool!
I'll also update with my breadboard, I think I had some things wrong.
Thanks for your help.