Hi and welcome.
The router will have a NAT firewall, which should be enabled by default.
This prevents any incoming traffic from the internet going straight to your computer unless it is something the computer has requested (i.e. when viewing a site etc) or unless you have created firewall rules designed to route traffic straight to a PC (used if you are running any server programs, games hosting etc).
So in that sense your computer is made safe from direct internet attacks.
What it is not safe from is any rogue programs which find their way onto your computer from a download, disc etc which might try and collect data about you and send it over the internet.
This is where ZoneAlarm is still useful - for controlling outgoing internet access on a program by program basis.
So really it would be wise to keep ZoneAlarm, it does do a good job. Secondly your router should be secure by default, with a NAT firewall enabled. If this is case (you can check by logging into the management interface) then you shouldn't really need to change anything else.