Later versions of Photoshop implemented what Adobe call JDI fixes, which stands for Just Do It. One of those fixes enable the saving directly to JPG, files with layers, or 16 bit, or with workpaths, alpha channels, or any feature that would otherwise need to be saved to PSD or TIFF. With CS5 you will need to flatten any layers, and make sure it is an 8 bit RGB color file.
Reading your question again, you might be asking how to make PS default to JPG when saving? If so, use Save As, and in the format drop down list, Ctrl click JPG. It will then use that format providing it is otherwise suitable (no layers etc.)
Yes, flattening the image first worked!
Much appreciated Trevor!
If I open a new file and go to save it the extension names show up (.jpg or .png).
That is not true. A net document save may default to .jpg. I do not ever remember a save defaulting to .png unless a the document was opened from a .png file. If the new document has layers save should open display the save as dialog and default to file type .psd.
However, if I open an illustrator file in Photoshop and try to save it the extension names are not there (it's just saving them as the proper format, but I want the files to have the .jpg or .png in the name without me typing them in myself.
I do not have or use Illustrator however I do not believe that Photoshop can open all illustrator file types like .ai. Photoshop may be able to import and export .ai file. If you have a document open in Photoshop you should be able to use "Save As" and chose the file type you want to save from the file type pull down menu. However .ai will not ne in the list.