It's really an upgrade that's necessary, not just an update.
The next OS X up from v10.5.8 is Snow Leopard.
Check the requirements here > Mac OS X v10.6 Snow Leopard - Technical Specifications
If your Mac meets those requirements, you can purchase Snow Leopard from Apple > Mac OS X 10.6 Snow Leopard - Apple Store (U.S.)
That will automatically update Safari for you but keep in mind, Snow Leopard is an older OS X. you may not be able to use all of Safari's features.
With Snow Leopard v10.6 installed, you may be able upgrade to OS X Lion. Requirements > OS X Lion - Technical Specifications
So the update for my current OS X system is Snow Leopard but that is actually old to the newest update of OS X Lion? How did I get so far behind on updates?!?! All I really want is to download a program like Word, on Apple I just found Pages. Can I download something that would be compatible even if I don't update my current version of Safari?? I clearly don't know too much about computers!! Thanks!
You are farther behind than you suspect. Since OS X Leopard 10.5 - 10.5.8 there have been;
OS X Snow Leopard 10.6 - 10.6.8
OS X Lion 10.7 - 10.7.5
OS X Mt Lion 10.8 - 10.8.5
OS X Mavericks 10.9 - 10.9.3 and counting
If you choose to upgrade from Leopard to Snow Leopard the retail disk that you by from Apple for $20 will bring your Mac to 10.6.3. You will want to update to 10.6.8 using the update combo below. This will give you the final public release version of the Mac App Store (MAS) with which you can buy many apps for your Mac, some of which are free. Unfortunately because you will still be running a legacy version of OS X, many MAS apps will not run on your Mac, they require a newer version of OS X. The version of Pages now sold in the MAS, for example, will only run on OS X Mavericks.
OS X Snow Leopard 10.6.8 v1.1 update combo
OS X Upgrades -
A Mac that can run OS X 10.6 Snow Leopard can usually run OS X 10.7 Lion. And a Mac that can run OS X 10.8 Mt Lion can also run OS X 10.9 Mavericks. Lion, Mt Lion and Mavericks can all be installed over OS X Snow Leopard 10.6.8, so you may upgrade from Snow Leopard directly to any of the other three. Both Lion and Mt Lion are paid upgrades, while Mavericks is a free upgrade. You may also upgrade directly to Mavericks from either Lion or Mt Lion.
System requirements for OS X Snow Leopard & OS X Lion
System requirements for OS X Mountain Lion
System requirements for OS X Mavericks
Be aware. If you upgrade to any of the Lion Family or Mavericks, any PPC apps or apps based on Rosetta cannot be used on these upgrades.
It means in effect that if you rely on the apps you currently use, upgrading will not only deprive you of their facilities but you will likely have to pay for replacements, and there is no guarantee they exist.
Before you upgrade, check out the apps you have with this link.
Application compatibility table - RoaringApps And if you want to investigate further, this excellent article from A Brody covers the matter in depth.