Start by checking if you can run Snow Leopard:
Requirements for OS X 10.6 'Snow Leopard'
http://support.apple.com/kb/SP575
(NB: PowerPC applications can still be run in Snow Leopard using Rosetta, but they will not work in later versions of OS X. For detailed information see this user tip: https://discussions.apple.com/docs/DOC-6271 )
The OS 10.6 Snow Leopard install DVD is still available for $19.99 from the Apple Store:
http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard
and in the UK:
http://store.apple.com/uk/product/MC573/mac-os-x-106-snow-leopard
Before installation you should back up all your data.
When you have installed it, run Software Update to download and install the latest updates for Snow Leopard to bring it up to 10.6.8, or download the combo update from here:
http://support.apple.com/kb/DL1399
Check via Software Update whether any further updates are required, particularly to iTunes (you will need the latest version in order to synch with mobile devices).
You should now see the App Store icon in iTunes, and you now need to set up your account:
http://support.apple.com/kb/HT4479
You can now upgrade to Yosemite OS 10.10 for free IF you have one of the following Macs, with not less than 2GB of RAM, and at least 8GB of available space on your hard drive:
Yosemite will run on the following Macs:
iMac (Mid-2007 or later)
MacBook (13-inch Aluminum, Late 2008), (13-inch, Early 2009 or later)
MacBook Pro (13-inch, Mid-2009 or later), (15-inch, Mid/Late 2007 or later), (17-inch, Late 2007 or later)
MacBook Air (Late 2008 or later)
Mac Mini (Early 2009 or later)
Mac Pro (Early 2008 or later)
Xserve (Early 2009)
The requirements for Yosemite are the same as they were for OS X 10.9 Mavericks,
iCloud system requirements:
http://support.apple.com/kb/ht4759
If you cannot run Yosemite you can purchase the code to use to download Mavericks from the App Store (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):
http://store.apple.com/us/product/D6106Z/A/os-x-Mavericks
or Mountain Lion:
http://store.apple.com/us/product/D6377Z/A/os-x-mountain-lion
Getting your Mac ready for a major update like Yosemite:
http://www.macworld.co.uk/how-to/mac-software/how-get-your-mac-ready-prepare-for -os-x-yosemite-release-how-to-download-yosemite-3521995/
Detailed review of Yosemite:
http://arstechnica.com/apple/2014/10/os-x-10-10/
