Yes, in deed you got proper stuff. To answer rest it will be litle long so by patiance.
In Photoshop in File is option Save for Web. Here you have 4 options of view Original. Optimized, 2-Up and 4-Up. Personally I prefer 4-up which will bring you Origimal plus 3 previews of your photo with different setings. On right side of this window you have options to change: GIF, JPG, PNG and WBMP.
GIF is very good with simple graphics( combination of lines, shapes of different colour) where colour is not gradient. This format is also able to create transparency on selected colour which you do not wish to see it.
JPG/JPEG is for gradient colour when you want it see changes from one colour to next one. So basicaly for photos and graphics which need gradual change of colour. Not able to work with transparency.
PNG in this days is some kind of hybrid between this to format. So you can save your graphic in this format if you wish. I use this format in a case of buttons when I use gradient colours but need transparency.
WBMP Is product of Microsoft some people use it over net but I dont like to.
For each format there are different settings to be change like for JPEG quality, blur. Preview window does show you which format you are using, how big it is gone be and how long this will take to dowload under certain conection speed.
What you have to do is simple made a changes to any of 3 preveiw windows to see difference. Once happy use Save button which will come up with options where to save it on your pc and that is done. (please dont forget highlight window with choisen setting otherwise you save wrong settings!!! Which is common begginer mistake.)

Hope this help you