Using something else than Winzip for a web page is IMHO a mistake. (I'm generaly upset when I have to deal with a compression tool I don't have) Winzip has become a standard, it's efficient and it's free. For your personal use on your computer, choose whatever you want anyway.
To reply to the question, I usually put all files and folders I want to zip in a folder(let's call it "X").
With the windows explorer, right click on "X" -> add to zip. Without checking any option, the sub-directories of "X" will be saved in the zip file. The main folder "X" is the root of the zipped tree, and it doesn't appear when unzipping the files.
Exemple : to pack the "starship" shipset :
