You will usually be storing your pictures in the camera on an SD or SDHC card. Just about any card reader will work with Linux, so no problems there. (Make sure your cards and card reader are compatible, though!) If the camera includes a cable for downloading directly to the PC, it will probably behave as a USB storage device, which again Linux supports. You shouldn't have any problems.
The software suite won't run on Linux directly, but there numerous photo manipulation tools available.