How did Pete manage to find Goofy and Max at the Neptune Inn? He wasn't following them; he left the campground before they did, and a long time and much activity had passed between the two events.
That's simple. They just ran into each other again. Pete's route could easily have been time-consuming enough that Goofy and Max could catch up, distance-wise. Pete did find what room they were in, but that's about it.
The location Max changed the map to was not mentioned at all during his conversation with PJ that Pete overheard, so how did Pete know that Max changed the map to LA specifically?
That's easy. While we can debate over who's the better father between Pete and Goofy it's obvious that Pete is more involved in the lives of his children than Goofy is. PJ only needed to mention the Powerline concert and then Pete could piece it together easily enough from there and he probably already knows all about Powerline because he cares what his kids are into if only to keep them from being involved with things he considers a "Bad Influence." I'd be willing to bet that PJ sneaks over to Max's house to play Grand Theft Auto.
PJ didn't mention the Powerline concert either, and there's no indication he talked to his father about it beforehand, and unless he mentioned Max going there to him, which he probably didn't or Pete would have told Goofy about it sooner, the fact that such a concert was going on still wouldn't tell Pete that's where Max changed the map to.
The scene cuts while Pete is still listening to Max and PJ talking. We don't see him hear Max mentioning LA later on in the conversation.
How many cups of sugar does it take to get to the moon?