My group is unsure how the scenario card 1B in Fog of the Barrow-downs, The House of Tom Bombadil is played. When I read it, it sounds like no one goes on a quest, but we still reveal encounter cards, travel etc. However others in the group say we have a round where we are completely free of any encounters.

Which is it?

http://hallofbeorn.com/LotR/Details/The-House-of-Tom-Bombadil-FotBD

You skip the whole quest phase, which includes sending characters on the quest and staging (revealing encounter cards). You still get to travel (theoretically, at least, if there were any locations in play) because that's the next phase.

Bombadil's house is like the calm before the storm!