Would it be worthwhile in any way for me to attempt mining on my Gateway Desktop?
Not unless you have a good ATI video card, or get electricity for free.
Without a GPU you will be losing money on electricity and it's painful how slow the BTC add up.
I mean I watched mine for a day and then gave up.
You need many GPUs to be making anything that you wouldn't be embarrassed to tell your friends.
They won't be impressed when you tell them you ran your computer for a week and made a whopping 26 cents.