It looks like I have Mac OS X version 10.4.11 with an Intel Core 2 Duo processor. When I click on software update it says that there are no updates available which I know is not true. I know that I need to update but at the low version I am at now I know I can't update to the Mountain Lion version. The next step seems to be the Leopard which does not seem to be available for under $200 and I would like to update fully to the Mountain Lion eventually so I don't want to have to pay that cost for something that I will be upgrading anyways.
I don't think you have an option but to find a copy of Leopard on eBay or Amazon.
Having said that, you will eventually have to get to 10.5.8 before you can upgrade further to Snow Leopard (10.6.3 and 10.6.8).
I very much doubt your machine can get as far as Mountain Lion.
I suspect you may have to buy that elusive (or expensive) copy of Leopard ... or progress to a more up to date machine, preloaded with Snow Leopard. If you take the latter option you can always consult through these columns as to whether the machine you intend to buy will meet the minimum specs to meet your aims.
Minimum specs for … Snow Leopard. Mac OS X v10.6 Snow Leopard - Read the Technical Specifications
Hope you find this helpful.