Your computer is almost certainly starving for RAM. 1GB was a very small amount of RAM 4 years ago, you should upgrade to the maximum your machine will support.
Check Activity Monitor (Applications - Utilities - Activity Monitor) when the computer is operating normally and see how much Free RAM it has. As a general rule of thumb if it has 500MB or less it's time for an upgrade. You will find the amount of Free RAM by clicking the System Memory tab in AM.
Your computer is almost certainly starving for RAM. 1GB was a very small amount of RAM 4 years ago, you should upgrade to the maximum your machine will support.
1GB is within spec for leopard, but I don't think SL or Lion would be fun on that.
Check activity monitor for ram. Check how your cpu cycles look there. If you're low on ram and incurring pageouts on an older hard drive that could do it. I'm thinking there's just as much of a chance of it being weighed down with flash or browser addons.
Hangs are going to happen on a Mac with only 1GB Memory. To minimize have nothing else open. 1 app at a time.
The random quits could be a corrupted preference file or a 3rd party extension.
Try moving your /Home/Library/Preferences/com.apple.safari.plist to the desktop and restart Safari. It'll create a new fresh .plist.
If you still have crashes move the .plist back.