First you will have to order Snow Leopard, http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard install that and on-line update to 10.6.8. Then depending on your Mac, see where you can go upgrading.
Lion requires an Intel Core 2 Duo or better processor, 2 GB ram, and 5 GB hard drive free space.
A more direct question, exactly what Mac do you have, what processor? If you click the apple left side of the menu bar, About This Mac it shows the processor.
If that says Core 2 Duo, i3, i5, i7 or Xeon you can move to at least Lion, 10.7.5. To do that you would have to contact Apple telephone sales (Lion is not in the store or retail store) at 1-800-MY-APPLE and pay the $20 fee for the download code...they send an email in a few days with the code and instructions.
If your Mac meets the requirements for Mountain Lion or Mavericks, found in http://www.apple.com/osx/specs you can do directly to Mavericks for free from Snow Leopard...skip the Mac OS X's in between.
With Snow Leopard installed, you get access to the Mac App Store for on-line software purchase and download.