Hi John..
first this is Linux, we dont have programs [unless your watching the TV] we have applications,
Now as you are using Mint there are 2 ways to safely remove applications, if it is something you have installed from a third party repository, then in mint menu under all applications search for what you wish to deleted, right click and select uninstall,
if it came as part of the Mint installation, the best way is to use the software manager, put the name of the app in the search bar, when you find it click on uninstall , in both cases unneeded dependencies are usually removed [but some 3rd party apps will leave a bit of crap around, to remove unwanted/unneeded desktop icons right click and select delete, [this will only remove the desk top icon and nothing else]