The format you use to save photos is really about how you work with the photos. Each has it's advantages to consider, and disadvantages.
JPEG: This is a "lossy" file format, but how much detail is lost is adjustable. It is a trade-off between file size and display quality. It is by far the most popular file format of non-professional photographers. What you should consider with this format is that if you adjust the compression the camera uses to save to high-quality, the final photos have enough detail to print with, even with some enlargement. The camera saves the file with it's own color table, white balance is fixed, the contrast and brightness levels are also set by the camera. You can edit some of this and adjust in software like photoshop or even free options like GIMP or Picasa. However when you save the edited file you will lose a bit more of the detail. Normally not too visible, but each re-save loses more detail. If you have no interest in adjusting and editing your photos with software prior to printing, there really is no need to use another format. Many cameras allow simple adjustments right on the camera without need to do this on a computer.
Raw: The image is saved with full detail. The white balance is not fixed, and you can correct it with software on a PC. Color balance and color space can be adjusted easily in post-processing. However the file sizes are comparatively huge which means you need a lot more storage space. You typically will export the file into a JPEG after making adjustments for display or possibly printing. TIFF is another option for printing. The Raw format is mostly supported by photo editors these days, was not always like that. Adobe Lightspace is the professional choice for Raw, I use RawTherapee as the free, open-source alternative.
TIFF: Can't be too specific here as TIFF is a very old format and comes with a lot of variations. It traditionally has been a format intending to be used to print with. It comes in versions that can be compressed in a "lossy" method or "lossless". It is not as universally supported for display on screens. Photo editing software pretty much supports the format as much as any other. But you have the same limitations as far as color space and white balance being set by the camera and not as easily corrected as Raw.
So it really boils down to deciding for yourself what your end product will be and what it will be used for. I personally do some small amount of printing, but most of my work stays as digital form to be displayed on computers. I save on camera as Raw + JPG (each image is save twice on the card, both formats). I mostly process my photos from the Raw images using RawTherapee. That gets 95% of the work done, then I export to GIMP (software uses TIFF as the intermediate format) for adding a copyright watermark and resizing the image to my final display size. I then export as a JPEG as my final product. Technically I use it all while working on a picture.
You can search photobucket for bfhammer to see some of my work.