Best OS X for iMac G3 500MHz?

If you upgrade it to 512mb or beyond (1GB is max), I'd recommend Tiger (10.4). Panther (10.3) would also work well as an OS, but not Jaguar (10.2) or below.

The reason I recommend Tiger over Panther is for software compatibility. Many new software releases now require Tiger or above. That includes Apple's Safari and the latest FireFox. This trend will continue, and Panther will become more and more abandoned. There are now some apps that require Leopard (10.5) and/or an Intel Mac, but for now, most developers still support Tiger.

Do you have an AirPort card installed for wireless, or are you using a third-party method?

Yes. Each +big cat+ release of Mac OS X is a separate major release. You need the full installation disk for Tiger, even if you already had Panther installed. There were actually NO upgrade discs sold. The only upgrade discs I know of were the ones included with Mac sold with Panther on the hard drive AFTER Tiger was released... a transitional free upgrade. You may find them sold on eBay, but you should avoid them and make sure it is a full retail installation disc. Retail is important; don't buy a disc that came with another Mac model. Retail discs are black with a big silver "X". It is usually on a DVD, so I hope your iMac has a DVD-capable optical drive.

It's good that you have already installed the 4.1.9 firmware update.

You can use these instructions for installing the AirPort card to see if you have one installed already.

http://support.apple.com/kb/HT2367

The only other wireless options are USB adapter and Ethernet Wireless Bridge. Check if you have an AirPort card first.

rccharles gave you some good links for various wireless alternatives.

The AirPort card from Apple that works in your iMac is 802.11b, which is the slowest of the common wireless protocols. However, the driver is part of the OS installation, in Mac OS 9 and Mac OS X, so it is likely to be the most compatible and easiest to use. The +Airport Extreme+ card will not work.

The USB wireless adapter requires a non-Apple driver, so make sure the product you buy includes a driver for Mac OS X (not just for Windows). It is much less likely the product will have a Mac OS 9 driver. Also, since your iMac only has the older USB 1.1 ports (not 2.0), you should confirm that the product will work properly over USB 1.1. Most that are sold now have at least 802.11g, which is much faster than the old AirPort card.

The Ethernet wireless bridge will look like a regular Ethernet wired connection to the iMac, so no driver is needed. Therefore, it should be compatible. It is probably also the most expensive option.

You can check the optical drive if you have a DVD movie. It may or may not play under Mac OS 9, but if your drive is DVD-capable, the DVD should appear on the iMac's Desktop.

For typical broadband connections, even the slow 802.11b of the AirPort card is usually not the bottleneck. The bottleneck on Internet connection speed is often the broadband connection itself. Other factors, such as the relatively slow (by today's standards) G3 chip, will also impact the maximum possible speed for Internet transfers. Therefore, if you can get a good price on an AirPort card (and adapter), you should not rule it out just because "g" or "n" is theoretically faster than "b."

For file transfer within your home network (from computer to computer), if the wireless router is at least 802.11g, then "g" or "n" are definitely faster than "b." The bottleneck in that case will be the AirPort card.

The AirPort Card will work with any standard wireless router. If that was not the case, a Mac laptop with an AirPort Card would not be very useful if you took it to a public wireless hotspot, like a coffee shop or the library. I've done that many times with my old 802.11b AirPort PowerBook.

If you are using Mac OS X on the Mac with the Airport Card, you can use either WEP and WPA security protocols.