Hi there,
I am safely driving an iMac 350 with Panther. It is running with 256 MB Ram which is enough to enjoy it but the more the better...
The best you can do is grab yourself an eBay Account, find some SDRam (100 or 133MHz) and a copy of 10.3. 10.3 is surely working and no longer sold by Apple and 10.4 is not suitable due to the 350 slot loader has no FireWire. It is a modern OS that works great. Get a full copy, no reinstall disc from eMac etc.
But there are some things to do before you upgrade to OSX.
First you have to bring your iMac from OS9.1 to OS 9.21 or OS 9.22 by installing this update:
http://download.info.apple.com/AppleSupport_Area/Apple_SoftwareUpdates/ (browse for your the language of your installed OS).
Then install the new Firmware. This is very important. The Firmware updater is only running on OS9.2 and you have to update if it is not already on Version 4.1.9. Look into System Profiler in the Apple menu for that information. If you can catch a copy of OSX 10.2 (Jaguar) the Firmware updater is on the discs (not on later versions!). Or ask a friend if you know one having that 10.2 CDs for help.
The Firmware can also be downloaded from Apples download site.
If you don't update and install OSX your iMac can decease when restarting due to a bug in certain FW versions. I was on the lucky side, it was working without because noone told me to do it before installing. I did it afterwards and the IMac is still working.