try surfing using the ip numbers instead,
for example, type:
and see the ip number,and use this in your browser.
also try other browser as "lynx", wich is included in slackware.
If you can surf as root but cannot as user i think <ry_Den> is correct and this is related with permissions in some file. Check if you have some hidden directory as ~/.mozilla or ~/.netscape in your home directory. If you dont have check the documentation of the browser.
I have this problem before but i cannot remember how was solved, but i hope this help. ^_^