Gene's literal journey is indeed to Leper's house as the two previous editors suggest.
However, Gene is taking a figurative journey in practicing being a true friend. He is practicing the skills of listening, trying to understand, trying to withhold judgment and we see Gene struggle with that a lot.
This is significant because Leper is laying the whole truth out on the line for him. Gene has opportunities to be honest with Finny (even though he tried once) and never is since that first time he tried. True friends tell the truth. This journey is significant because Gene watched a friend reveal the truth and he was in the situation of accepting the truth... the same situation that Finny will later be put in.
In A Separate Peace, Gene's journey to see Leper results in the destruction of still another illusion the boys at Devon use to cope with their situation. Leper's insanity and desertion from the army destroy the lighthearted myth of Leper as war hero gallivanting around the globe helping the Allies win the war.
Significatly, though, in his insanity Leper still has moments of lucidity during which his mind is clear and somewhat rational. And he can still remember certain details clearly. These details, which include the moment just before Finny falls from the tree, will prove vital later in the story at the mock trial the boys hold in order to find out what really made Finny fall.
In chapter 10 of "A Separate Peace" Gene is contacted by Leper and asked to go to his parent's Christmas location. He travels to his home where he finds a mentally disturbed Leper. Leper tells Gene he has gone AWOL from the military. He did this because he believes he was getting a discharge for mental instability.
While at the home, Leper confronts Gene about having Gene causing Finny's fall from the tree. Gene gets angry and kicks the chair out from under Leper causing his mother to run into the room. She later forgives him for his actions over dinner.
Following dinner the twoboys go off walking. Leper desperately needs to talk to a peer and starts crying and talking out of his head. Gene can not stand the situation any longer and leaves Leper in the woods and returns to school upset.