Hey Candi, if you do a topic search, there have been other discussions on packing. But I can quickly answer your questions-
The bathrooms have a shampoo and soap (shower gel) dispenser. If you are picky about your shampoo, then bring your own. We normally just bring a "scrubby" and use the shower gel provided. Hairdryer- yes, iron- no -only in the little laundry rooms. If you need something pressed, the onboard services can do it for you. Alarm clocks- no, but they have an automated system to set a wake up call that works very well so you don't need one.
Beach towels are not HUGE but they are a good size. Don't bother packing towels unless you have lots of extra space. They provide clean ones if you leave yours on the bath floor for the cabin steward.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007