What is the Franck-Condon region
The Franck–Condon region is a region of the final potential energy surface which is accessed by the vertical transition from the initial state. Usually, the initial state is the ground vibrational state of the ground electronic state and the final is some vibrational state of an electronically excited state. For instance, the Franck–Condon region is shaded below on the Figure from Wikipedia article on the Franck–Condon principle.




  But when a transition takes place from S0 to S1, it is said that
  molecule is excited to the FC region of the S1 state - but there is no
  guarantee that going vertically from an S0 minimum will lead to a near
  S1 minimum. So something is probably incorrect with my current
  understanding.


Nope, what you are saying is correct. Indeed, according to the Franck–Condon principle vertical transitions are the most probable. The minimum of the upper potential energy surface might lie outside of the Franck–Condon region, so a vertical transition from the ground vibrational state of the ground electronic state to the excited electronic state might indeed result in one of the vibrationally excited states like it is shown in the picture above. Once a molecule is excited in such way it quickly relaxes to the lowest vibrational level of the electronically excited state from which it can decay then back to the electronic ground state. And again, on this way back to the ground state the most probable transition is the vertical one, so that photon emission can lead to a vibrationally excited state of the ground electronic state, not necessarily to the ground vibrational state of it.