For "shortcuts", drag the icon from the Applications folder to the dock (taskbar), not the desktop. It will stay, even after you quit the application. If you hold the mouse button while over a dock icon you will see an option to keep or remove it from the dock. Dock items are just aliases. The application is still in its original place. Dragging to the desktop actually moves the file. Adobe will not stand for this, and does its best to restore order to the universe. Unfortunately, its best is not always good enough. If you really want the shortcut on the desktop, not the dock, hold down Command and Option while dragging it. This will generate an alias, and leave the original file unmoved.
I wanted to go online and could not find the Firefox icon, after searching the computer the only thing I found was the downloaded install file?
When you downloaded Firefox, it was in a .dmg (disk image) file. You must have been running it from there. Double-click the .dmg file to mount it, then drag Firefox from the .dmg file into the Applications folder, and from there to the dock. You can then dismount (eject) the disk image file and delete it.
You should set up a non-administrator account for normal use, and reserve your administrator account for software installation and system maintenance. This gives the system some protection from malware, and from you. For example, if you drag an application to the desktop, thinking you are making a shortcut, then later, delete the supposed shortcut; the application is gone. If you were running as non-administrator, it would not let you drag from the Applications folder. You could still drag to the dock, or Command-Option drag aliases.