You say you are new to web design, but are you new to design? I ask because if you are speaking in terms of something like Photoshpo with 8 bits per channel, then yes 8 bits is more than sufficient. More than that is only needed for print mediums.
When discussing file formats, I would typically stick to PNG or JPG depending on the nature of the image and whether or not you need transparency. For instance PNG supports transparency in images whereas JPG files do not.
When you use SAVE FOR WEB in Photoshop or Illustrator, all images are saved as 8 bit, that's because it is impossible to use 16 bit images on the web. If anyone doesn't believe me, try using a 16 bit image. It will not work.
As for image type, I try to use GIF as much as possible, for clean graphical elements. You have good control and the images can be very small.Of course it is not the type of choice for photographs.