How to Wire Fan with Black/White/Green to Ceiling with Black/White/Red/Bare?
In some installations, there will be two switches for a fan - one for the fan and another for the light.  The ceiling fan would have four wires (blue is common color for the 4th wire), and they are normally labeled Light, Fan, Common, and Ground.  One wall switch will control the black wire, and the second switch will control red.

Your new fan either doesn't have lights, or the lights can't be controlled by a switch independently from the fan.  The documentation should tell you what each wire is for, but your assumption should be correct - black to black (assuming your switch controls the black wire - check it), white to white, and green to ground.  The red wire should simply be capped off by itself in the junction box.