The Tower of the Elephant and Heretics of Tarantia are two great adventures, IMO.
Tales of the Black Kingdom is sort of a mini-campaign comprised of three loosely connected adventures (I think we played through the whole thing in roughly six sessions). It has some minor quirks, but there are a couple of threads on these forums that discuss ways to fix them. Overall it is good stuff with a nice Conan-vibe, and we had fun with it.
If you want to quickly get a campaign started, I would recommend you to just get those three books and choose from them what you'd like to run.
There is other stuff as well, though:
Reavers of the Vilayet can probably be a fun adventure, although from reading it there were a few things that felt a bit odd to me. I'll probably try to use it (slightly modified) in the future, though.
The Lurking Terror of Nahab I personally didn't like all that much.
Across the Thunder River has a campaign included (it's about 30 pages, IIRC) that I think could be a lot of fun (I'm thinking of using it for my next Conan game). It's more of an campaign outline, though, and will probably require a little more preparatory work than what you're looking for right now.
Messantia has a 32-page adventure included that was quite decent.
Shadizar likewise includes a 32-page adventure, but this one I wasn't too fond of.
The Conan Compendium includes a number of shorter adventures (was it three?) that I haven't really looked all that closely at.
If you don't mind putting in some extra work, you could probably get a lot of cool stuff out of using the adventure locations in Ruins of Hyboria and the original Conan tales (Ruins doesn't have any ready-to-run adventures, though). It's probably not what you're looking for, but Ruins is such a fun book that I feel a need to recommend it anyways. :wink:
Oh, there are also two PDF-only adventures; The Coming of Hanuman and The Black Stones of Kovag-Re. Hanuman is OK, Black Stones not so much.