The topic title is the basic question, but here's my over-analysis:
Under "Encounters in the Catacombs" on pg.21, it states "a Hero in the Catacombs must draw a Catacomb card at the start of each of his turns." (emphasis mine.)
Since the player has already started his turn and done any "start of turn" effects before choosing to search, move, or enter the catacombs, I figured that a player would not draw a Catacomb card the turn he entered it. He would just swap his figure with his travel marker and that would be it.
"Start of Turn Effects" even have specific clarifications on pg. 27. The second paragraph shows that drawing a Catacomb card is clearly a "start of turn" effect (but it happens after the others are resolved.)
I think this supports my reasoning that you don't draw a Catacomb card the turn you enter them. That said, I hope my interpretation is wrong. Playing that way doesn't feel right and it's difficult to explain to a person why they don't draw a card that move. Any other action results in a card or tile: Move: Draw a tile and/or a Dungeon card. Search: draw a Search card. Enter Catacombs: no card this turn (?)
So, do I draw a Catacomb card the turn I enter the catacombs? I hope the answer is yes. But "yes" may need errata/clarification support.