Q: Browser Upgrade

We can't help you much without more information from you. For any upgrade question we need to know:

1) You Mac model and sub-model

2) Your current Mac OSX version

     [both are available from "About this Mac in your Apple menu.]

3) What browser and its version are you running now.

Without that information we are guessing and you really don't want people here guessing. That can have a bad outcome! Think of coming here as being like phoning an auto mechanic who does not know you and saying, 'My transmission isn't working right." The first thing the mechanic is going to ask is "Make? Model? Year?" It's like that here.

Also, you found a litte-viewed forum for Apple hardware made before 1998. Not your fault because Apple chose a badly non-intuituive name for the section. However, being here is like asking for help in a  tomb--very few Helpers visit here.

Once we know your equipment we can ask the Host to move you to a more active and appropriate forum where you can get much faster help.

iMac 8,1 at 2.4 GHz sounds like this model

http://www.everymac.com/systems/apple/imac/specs/imac-core-2-duo-2.4-20-inch-alu minum-early-2008-penryn-specs.html

You can upgrade its RAM to 4GB ("official" max).  Its RAM slots are accessible from bottom edge of iMac.  There are two slots.  RAM modules are available from sources like Crucial and OWC.  Here's a list of options from OWC for your iMac (if the above model is correct)

https://eshop.macsales.com/upgrades/imac-20-inch-early-2008-2.4-ghz/memory

OWC has installation How To videos (linked to product description).  So, if you want to upgrade its RAM, you can upgrade its OS and software to as high as El Capitan (11.11.x).  Since you are currently at Leopard (10.5.8), you first need to upgrade it to Snow Leopard (10.6.8), by purchasing the Snow Leopard installation disc, available for $20

http://www.apple.com/shop/product/MC573Z/A/mac-os-x-106-snow-leopard

Snow Leopard gives you the Mac App Store, needed to download and install El Capitan (free upgrade).  If web browser compatibility is your main consideration, El Capitan gives you the latest version of Safari, and access to the latest versions of other web browsers (like FireFox).

NOTE:  You should back up your system (at least your valuable user data) before applying a major upgrade.  You can use the built-in Time Machine feature (to back up to an external drive), if you do not already.  Also, some of your third-party software currently being used may not be compatible with the upgraded OS.  For example, Leopard (10.5.8) can run applications coded for older PowerPC Macs.  Mac OS X releases after Snow Leopard (10.6.8) cannot.

Thanks! Getting you moved to the forum for current iMacs, of which yours is one.

Forgot to include link for El Capitan upgrade information

Upgrade to OS X El Capitan - Apple Support

NOTE:  El Capitan is almost the latest OS release.  macOS Sierra (10.12) was release a few days ago, but your iMac model does not meet its system requirements.