Q: Best OS system?

I think it is OS X Mt Lion. But you need at least 2GBs of RAM and even then it might run slow. 4GBs is better for any of the newer Download only versions of OS X as they all use a lot more system resources than the version you are now running.

Snow Leopard is probably your best OS.   Without more details it is not possible to say if you can go further but my guess is that you can go to both Lion family and Mavericks

Check your apps first.   PPC apps do not work on higher OS versions than 10.6.8. ... and upgrade your RAM to 4GB if you can.

I go with Snow Leopard.   In my view it is the best.

Sorry, forgot to put these links in for you ... Read these spec sheets to get a clearer idea of what your machine can take.

Mac OS X v10.6 Snow Leopard -Read the Technical Specifications

And for Mountain Lion

technical specification

For what reason do you want to upgrade your OS X version? Is there something Leopard doesn't do for you that one of the newer versions will do?

If you are using some older PowerPC type programs those won't run on Lion, 10.7.x, and above as Apple removed all ability to run those older program in the newer download only OS X version. They still work under Snow Leopard.

It is thought by many that Snow Leopard was the best OS Apple even made and I agree in some ways. But if you need the iCloud stuff then you need a newer OS X version than Snow Leopard.

Here are the specs for your machine:

http://www.everymac.com/systems/apple/imac/specs/imac-core-2-duo-2.4-20-inch-alu minum-specs.html

Even though it appears it can run the current/latest OS, based on the age and RAM limits, I'd choose Snow Leopard and get at least 2 GB of RAM (or the max of 4).

Not sure why a lot of Apple users think that upgrading the OS will fix problems with slow apps or other problems that are most likely related to a hardware issue.

If you are finding your system is running slower then usual and some programs don't react like they use to upgrading the OS will not help in any way and more than likely only make it worse. Since you are using a older Mac your problem could be caused by the hard drive starting to fail or some other hardware problem. Or it could be your hard drive is nearly full which will slow OS X down to a crawl.

That may be a guide.   Chrome is not totally well regarded around here.   I hope you haven't been tempted to download lots of extentions or security aids.  They have been known to interfere with each other and other apps on a Mac.

Hoping too you have not been tempted to download other 'security' aids, Norton and the like.

Chris Berry wrote:

My machine's having trouble with Google Chrome and Safari is running very slowly. I thought upgrading the OS might help as can't afford a new machine at the moment?

See:

Mac Maintenance Quick Assist,
Mac OS X speed FAQ,
Speeding up Macs,
How to Speed up Macs,
Macintosh OS X Routine Maintenance,
Mac troubleshooting: What to do when your computer is too slow,
Essential Mac Maintenance: Get set up,
Essential Mac Maintenance: Rev up your routines,
Maintaining OS X, 
Five Mac maintenance myths and
Myths of required versus not required maintenance for Mac OS X for information.