Note that Firefox will completely lock up when browsing many websites on your PC with only 512MB RAM. That's not a Tiny Core issue, modern Firefox needs more RAM than that for most usage regardless of the OS.
TC has extensions for other Web browsers that work better with low RAM. Your internet is working fine, so use the apps browser to find and install them via a GUI as described in the book. Seamonkey is based on older Firefox with lower system requirements, and there's also netsurf, dillo, links-full...