Why won't the internet (wifi) work?

It could be any number of problems. OS 10.2 is obsolete and does not support modern wireless encryption, so it may not be working with the security settings you have in place on your router. As a test, you can disable wireless security on your router, and if it suddenly works, you know that's the problem. Even with Tiger 10.4 (the OS that you should get and install on your machine), due to the wireless-b card your machine uses, you cannot use WPA2 encryption, which is the standard on a lot of newer routers, so it's necessary to re-configure them to use WPA Personal instead. I don't recall what encryption 10.2 is compatible with, but it's probably WEP, which is not all that secure and should be avoided.

Internet Explorer is also obsolete and unsupported on the Mac at this point, and even if you get it working, it's going to be a sad browsing experience, with the vast majority of websites not coming across correctly, if at all. There may be an old version of Firefox you could get and install on 10.2, but really any effort you put forth should be spent getting 10.4 Tiger on your machine. I'd try the wireless security test I mentioned above first, but then definitely go get Tiger.