Update FF at your own risk, FF 3.6 is relatively stable with Tiger, I wouldn't attempt to update it further with your OS.
If your machine has a dual processer (intel), you can upgrade the OS to Leopard, perhaps Snow Leopard. Then you have more browser update options.
If your machine is a PPC, then OS Tiger (10.4) is the end of road.
Camino browser is a Mozilla product, as is Firefox. It's Mac native, and is not a hog like Firefox. Why not try it? The interface is simple, you can import bookmarks easily, and the current version is OS 10.4 friendly. I use it occasionally to access Ancestry (troubleshooting) and it's fine.