Since you are new to the alternate browser scene, I should explain to you the choices:
If you have a Linx of OSX browser, Safari and Konqueror, based on the KHTML rendering engine, are very speedy but have a couple kinks in the rendering. A very good choice if speed is key.
Opera became popular during the time Netscape was rebuilding their browser (i.e. creating Mozilla). They invented tabbed browsing and have a good, stable system with a skinnable UI. The only downside is that you have to pay for it.
Mozilla, which is a less bloated Netscape 7.x (RIP) is an open source, cross platform project that was meant to be the developer of Netscape. With the demise of Netscape, Mozilla it now an end-user product. It is big, but full featured, with tabbed browsing, pop-up blocking, a competant mail client, skinnable, an IRC client, a composer, and you have the option of getting extensions for even more features. The rendering engine is the most standard-compliant out there. Also, the fact that it is open source led to the creation of offshoots like Galleon, Epiphany, Phoenix, Minotaur, and Camino.
After releasing Mozilla 1.3, the developers decided that Mozilla was getting too bulky. The Phoenix Project, a slimmed-down Mozilla with just the browser, was chosen to be the successor to the Mozilla browser (it is now called MozillaFirebird due to copyright issues). Thunderbird (formerly Minotaur) will replace the mail client as a stand-alone program. The final version of Mozilla 1.5 on will be based on Firebird. Firebird, while speedy and small (1/3 the file size), is in beta form and does not have all the bugs out of it.
As long as you don't mind the occasional stability issues, Firebird is the best. If you want a rock-hard browser, try Mozilla or Opera if you are willing to pay for it.
Note: This post was written on Firebird, and I have Mozilla on my system as well, so I am somewhat biased here. 🙂