Is the firewall in the wireless router enough?

I got an ADSL internet connection that goes into a Linksys wireless router that features a built-in firewall. Are those kinds of firewalls typically good enough for regular use, or should look into securing the computers on our home network more tightly?

I generally use a hardware and a software firewall. Hardware firewalls are great at blocking all incoming connections, except for those allowed. However, they do a very poor job at trying to decipher traffic going out. There are often many applications that will connect through port 80 going out, which should be entirely allowed by the hardware firewall. A software firewall here will help decipher which of those programs should be allowed to go out, which will help prevent botnets/trojans/other virii from phoning home, so long as you keep your firewall up-to-date. Because you can easily give yourself malware with a good hardware firewall in place, a software firewall is necessary to catch some of the "Whoops" factor.

it will protect you from outside attacks reasonably well.
It doesn't protect you from something on another computer on your network, it also wont tell you if you have something nasty that is calling out - for these you need local software firewalls.

Well, 90% of the problems with wireless routers are leaving it open (such as using WEP), or using weak encryption. If someone can get on to the network, they can read all the unencrypted internet traffic using certain DHCP attacks. It is, however, very important to look online for security vulnerabilities for the router you are using, and make sure that you get the latest security updates.

Other than that, though, the default firewall and the firewalls on the computers should suffice. If you're particularly concerned about security, try installing Tomato on your router, and then using the firewall in that. I've heard very good things about it.