Assuming you can run it:
Start by checking if you can run Snow Leopard:
Requirements for OS X 10.6 'Snow Leopard'
The OS 10.6 Snow Leopard install DVD is available for $19.99 from the Apple Store:
and in the UK:
but nobody knows for how long it will be available.
When you have installed it, run Software Update to download and install the latest updates for Snow Leopard.
You should now see the App Store icon, and you now need to set up your account:
To use iCloud you have to upgrade at least to Lion, but some functions are only available in Mountain Lion:
You can also purchase the code to use to download Lion (Lion requires an Intel-based Mac with a Core 2 Duo, i3, i5, i7 or Xeon processor and 2GB of RAM, running the latest version of Snow Leopard), or you can purchase Mountain Lion from the App Store - if you can run that:
You may be able to upgrade directly to Mountain Lion; as I cannot tell the age of your machine I cannot be sure.
But you can check it out.
Before buying anything you should first make sure your machine will accept the upgrade you have in mind.
Minimum specs for Snow Leopard. Mac OS X v10.6 Snow Leopard -Read the Technical Specifications
Before you start upgrading, back up your files.
Once you have upgraded to Snow Leopard you will need to apply the Combo updater. Mac OS X 10.6.8 Update Combo v1.1 and follow with a software update to pick up any later updates.
Upgrading past Snow Leopard.
For Lion, the minimum specs are …
ownload code has to be bought from Apple.
For Mountain Lion, minimum specs …
technical specification Check before buying anything.
You can go directly from Snow Leopard to Mountain Lion via the App Store, providing you meet the specs.
If you can use their online version of turbotax software, and your browser can handle the webpage requirements of their online version of that software, that is an option that should not require an OS X upgrade.
To upgrade to a newer OS X version could require a hardware upgrade (new computer purchase) if your Mac is a modern antique. You may be able to update a browser or may find items such as Flash plugins lacking.
See turbotax support to find what they say about their software limits:
Note the hiearchial path to specific version requirements, it may help.
The offline actual install CD/DVD would require you to go online to download recent updates, btw. Depending on where you are in the world, the online version may not be available to you; so you'd need the hardcopy. I'd be looking at Apple's refurbished Macs if I were going to get an OS X upgrade.
Good luck & happy computing!
PS: Time-out occurred limiting an update to my earlier post... edit to add:
TurboTax 2009-2012 System Requirements
This information will tell you what each version requirements are and that
goes back to Turbotax 2009 when Tiger 10.4.x was among the supported.
Note the other information where answers may be hiding in their site.
Hopefully this helps you or another reader here.
Good luck & happy computing!