Ubuntu will support anything you need.
You can use gwenview (free, for KDE) or gqview (free, Gnome), ifranview on WIine, etc.. There's undoubtedly an image viewer pre-installed with your distro and there are many others available in the repository. Most also allow basic editing, such as cropping.
VLC (free) will play your video clips.
If you plan on shooting still images in RAW instead of JPG, there's rawstudio to convert your files into something manageable.
For advanced editing, there's GIMP (free) or you can use any recent version of Photoshop ($$$) in combination with Wine (free).
To transfer files to the computer, just pop the camera's memory card into the computer's built-in SD slot. If your computer doesn't have one, or if you plan on buying a dSLR that takes CF cards, invest $10 in a card reader.
The software disk included by camera manufacturers simply does some or all of the above with proprietary software. Sometimes the disk does include a Linux version. Most of the time the above software does a better job.