You need to be running v10.6.6 at the least to have access to the App Store.
Click your Apple menu icon top left in yourr screen. From the drop down menu click: Aboout This Mac
What version does it say?
If you are running v10.6.6 or higher, you can upgrade to Lion now but make sure your Mac meets Lion's requiremens.
- Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7, or Xeon processor
- 2GB of memory
- OS X v10.6.6 or later (v10.6.8 recommended)
- 7GB of available space
Mountain Lion won't be released to the public until sometime this summer and at this time, the requirements are unknown > Apple - OS X Mountain Lion