When a program tries to access the Internet, ZoneAlarm should pop up a little message box. There should be a check box you can tick which says something like 'Remember this program'. You shouldn't then be asked if that program can access the Internet again.
 
Otherwise, goto the Programs tab on ZoneAlarm, right click the program that you want to access the net, and choose 'Allow' in the Internet section of the menu that pops up.

get rid of zone alarm and get a router

That's fine and dandy if all you want to do is stop inbound stuff, but if you want to stop outbound (read: trojans) you need something like Zone Alarm.
 
FWIW, I've had trouble with ZA and XP but there's an update (2.6.362) out that is working mostly OK for me. I still get an error message in event viewer but ZA is functioning like it should. Tested with "Leak test" for outbound, stuck my head out from behind my router and tested ports with DSL Reports security test, and HackerWacker.

Well i will just try to install it again. Lets hope it will work now. Thanks for helping me.Lucifix