Why the frequency changes
As a satellite comes towards you, its signal arrives at a higher frequency, and as it goes away, a lower one, the Doppler effect. The shift is the frequency times the satellite's speed towards you divided by the speed of light. For a low satellite, the speed towards you peaks near the horizon at about 7 km/s and passes through zero at the closest point. That's about ±3.5 kHz on 2m and ±10 kHz on 70cm. Your uplink is shifted too, so you transmit slightly lower than nominal at the start of a pass and higher at the end.
The practical way
For FM satellites, program five memories on the higher band, usually the 70cm downlink, at +10, +5, 0, −5 and −10 kHz, and step through them during the pass, leaving the 2m side alone. The 2m shift is small enough that the satellite's receiver tolerates it. For linear transponders and SSB, use software such as SatPC32 or Gpredict with CAT control to track the shift automatically.
The satellite pass predictor gives the times and frequencies of real passes, and the live tracker shows where the satellite is.
Questions
How much Doppler is there on 70cm?
Up to about ±10 kHz for low satellites like the ISS and SO-50.
How much Doppler is there on 2m?
Up to about ±3.5 kHz, small enough to ignore on FM uplinks.
Which frequency should I correct?
The higher one, usually the 70cm downlink. This is often called the 1/N rule.
Is Doppler zero at the highest point?
Yes. At the closest point of the pass the satellite isn't moving towards or away from you, so the shift passes through zero.