Yes, you should definitely backup your data. And, before doing that you should repair your hard drive and permissions.
Repair the Hard Drive and Permissions
Boot from your Snow Leopard Installer disc. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities menu. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the installer.
If DU reports errors it cannot fix, then you will need Disk Warrior and/or Tech Tool Pro to repair the drive. If you don't have either of them or if neither of them can fix the drive, then you will need to reformat the drive and reinstall OS X.
After installing Snow Leopard download and install the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8 and give you access to the App Store.
Then should you wish to go further:
Upgrading from Snow Leopard to Lion or Mountain Lion
You can upgrade to Mountain Lion from Lion or directly from Snow Leopard. Mountain Lion can be downloaded from the Mac App Store for $19.99.
If you sign into the App Store and try to purchase Mountain Lion but the App Store says your computer is not compatible then you may still be able to upgrade to Lion per the following information.
A. Upgrading to Mountain Lion
To upgrade to Mountain Lion you must have Snow Leopard 10.6.8 or Lion installed. Purchase and download Mountain Lion from the App Store. Sign in using your Apple ID. Mountain Lion is $19.99 plus tax. The file is quite large, over 4 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.
Macs that can be upgraded to OS X Mountain Lion
1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later
2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later
3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later
4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later
5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later
6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later
7. Xserve (Early 2009) - Model Identifier 3,1 or later
To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.
Are my applications compatible?
For a complete How-To introduction from Apple see Upgrade to OS X Mountain Lion.
B. Upgrading to Lion
If your computer does not meet the requirements to install Mountain Lion, it may still meet the requirements to install Lion.
You can purchase Lion by contacting Customer Service: Contacting Apple for support and service - this includes international calling numbers. The cost is $19.99 (as it was before) plus tax. It's a download. You will get an email containing a redemption code that you then use at the Mac App Store to download Lion. Save a copy of that installer to your Downloads folder because the installer deletes itself at the end of the installation.
Lion System Requirements
1. Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7,
or Xeon processor
2. 2GB of memory
3. OS X v10.6.6 or later (v10.6.8 recommended)
4. 7GB of available space
5. Some features require an Apple ID; terms apply.