I have never personally noticed that if I just open an image and close it, without doing anything else, that it loses quality. However, I've read that this is the case.
Of course any image will lose some clarity if taken from a larger size down to a smaller size. And, it's impossible to make a .jpg file larger simply by resizing and still maintain good quality.
One of the things you need to be concerned with in .jpgs is the compression level. Often people compress too much resulting in too much image degration. Once you've lost that quality you can never recapture it without rescanning or saving again from the original image which is not "lossy" (such as a .tif file).
.jpg is a "lossy" type of file compression. Pixels are actually thrown out during the compression. If the quality level of the compression chosen is high, this will never be seen with the naked eye.
Below are 4 versions of the same image. Top left is the original image. On the right is the same image with a little compression, which still renders high quality. You can see some slight differences, but the quality is still very good.
Bottom left is a medium level of compression. Notice how the quality is starting to degrade. On the right is a high level of compression. The quality is now severely degraded.
The level of compression is an option at the time you save a file as .jpg. Look for it when you do the save.
Be aware that .jpg and .gif files are all that's recognized consistently on web pages. .png is an up and coming file format, but is not recognized yet by all browsers.
Here's some information you can read, which explains better than I have. If you focus on .tiff (or .tif), .jpg (or .jpeg) and .gif, you'll minimize any confusion.
http://www.scantips.com/basics09.html