|Give it a try. 10.53 is the latest release but if you want to try 10.60 Beta, then go for it.|
To find out how much memory you have, right click on my computer and click on properties. The amount of RAM will be listed on the first screen.
If you have insufficient memory, none of the commonly used browsers will work a miracle for you. For example, IE7 would be a better choice than IE8.
As far as Chrome, Opera, FireFox, etc. you'll just need to try 'em and see what you like best.
Here's some info on memory usage, speed, etc.