Any ideas? I have all the stuff from the kickstarter.
If you want to run published adventure, I would recommend to start with Pits of Kutallu. It is quite interesting one, monsters, cultists, slavers and shipwrecked at lost islands. The noble lady that the characters need to rescue is also a great hook for further adventures. Perfect start of a campaign, like I did.
If you don’t want to deal with complications of getting the players off the island, you can start with Vultures of Shem from the core book, which is also great - combat, mystery, some connection with Stygian nobles and fight with undead-like creatures.
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I ran The Tower of the Elephant, which is an adventure for the old Conan d20 game. You would have some conversions to do but story-wise it was really cool, especially for players (and GMs) that are new to the Conan world. It basically gave a brief overview of the timeline. Also, there are some choices to be made that will establish whether it is a good or an evil party (or something in between).
This answer might not be helpful to you but I thought I’d mention it. I have not touched the adventure book. I agree with @Valyar as I ran both the Pit of Kutallu and Vultures of Shem as playtest adventures and they were very nice and well received by the players, although the story is very similar.
Thank you for the suggestions 
I ran The Ghost of Thunder River a couple of times at UK Games Expo, and it worked well enough to bring new players aboard and get people into the swing of things.
Since this adventure includes a prelude, where players take for a short scene the roles of Pict warriors, did you use this during the expo session? I am considering to have this as an option for introductory adventure in the future, but this might be a bit confusing for new or inexperienced players or waste time from the window you have.
I skipped the prelude both times, and the game fit into a single 4-hour session nicely on both instances.