Q: Need help with connecting Power Book G4 to wireless internet

The icon in the menu bar can be hidden or displayed.  The setting is on System Preferences Network pane.  Where it says Show, select AirPort.  There's a checkbox for Show AirPort status in menu bar.  From that AirPort icon in menu bar, you can select the a local wireless network to join.

From that System Preferences pane, you can use more specific configuration settings, although the default settings often work unless you use special settings with your wireless router.

The AirPort Extreme card may be for 802.11b and g, and not n.  If your wireless network is set up for 802.11n-only, you'll need to change that setting to allow 802.11g (on the wireless router).

Well, it works well with Apple's AirPort basestations, but the PowerBook wouldn't be that useful if you couldn't take it the local café and connect to their "generic" network...

You do need a wireless router.  But it can be any standard wireless router, even a really old one.  A "modem" is something that is used to connect your computer (or network) to the Internet, through your service provider.  In the old days, they connected through the phone line and made annoying sounds.  Most users have a "broadband" connection these days, and you often "rent" the modem from your service provider.  Some more recent ones have a built in wireless router.  Do you have a more recent Mac or Windows PC that connects wirelessly?  How does it connect to the Internet?

When you click on the AirPort icon in the menu bar, do you see a list of nearby wireless networks?  (First, make sure AirPort is turned ON.)  Do you see the one you use with your newer computer?  If you do, what happens when you select it?

could I use a USB wireless adapter instead?

You could, but you would still need to connect to your wireless network with it, which is basically the same way you connect with your built-in AirPort card.  It's actually more complicated with a USB wireless adapter, because you have to install drivers and software to make it work, and that software is often not compatible with your old version of Mac OS X.

If you did not change any default settings on the router, it is probably already set to support the older (slower) protocols, to be most compatible.  You'll need to look up the manual for your particular router online.  Apple provides an app (called AirPort Utility) for managing Apple's base stations, but most routers use a web browser to access and change the router's settings.

after I enter the password but then when I try to open Safari it won't open.

Safari should "open" a window, even if you don't have an Internet connection.  Are you saying you don't get a Safari window at all?

Since you changed some network settings, you may want to restart the Mac.  Connect to your network before restarting.

Also, as a test, if you have a regular Ethernet cable, most routers also have connection points for a wired network connection.  You can try connecting the PowerBook using  a wired connection.  If you manually turn OFF AirPort, I think it will automatically use the wired Ethernet connection.  See if you can do some web browsing.  If Safari is your indicator of an Internet connection, you should confirm Safari is working properly.  If something is wrong with your system, and Safari is not working properly, it won't serve as a good test to see if your wireless connection is working or not.

Good job...   As a final piece of advice for new users of old PowerPC Macs, download TenFourFox. 

http://www.floodgap.com/software/tenfourfox/

It is a build of FireFox optimized for PowerPC Mac, and maintain by a dedicated developer.  Your version of Safari (even after the update you installed) has not been updated by Apple for MANY years, so it often does not work properly on some web sites of today's Internet, and its security is old.  I only use TenFourFox on my old PowerBook (which is older than yours).