Reinstalling would be overkill, and it still might not resolve the real issue.

Whether you opt to use Konqueror, Firefox, or any other web browser will not affect your internet connection. There are a few things you need to check to ensure that you can connect to the Internet. However, before I get into that, I need to know whether you connect to the Internet via your router or via your computer. From your problem description, it kind of sounds like you connect to the Internet via your router.

Assuming that you do connect to the net via your router, I'll proceed with some of the other checks that you'll need to perform to ensure that you can access your internet connection via Linux.

You need to check your network connection between your Linux box and your router. Do you know the IP address and netmask of your router? If not, then you'll need to obtain them from the documentation of your router. Let's assume that the IP address and netmask of your router are 192.168.0.1 and 255.255.255.0 respectively. Check to see if your Linux box and router can talk by typing the following (via Konsole): ping -c 1 192.168.0.1. You should receive something like this:
PING 192.168.0.1 (192.168.0.1): 56 data bytes
64 bytes from 192.168.0.1: icmp_seq=0 ttl=64 time=0.6 ms

--- 10.0.0.7 ping statistics ---
1 packets transmitted, 1 packets received, 0% packet loss
round-trip min/avg/max = 0.6/0.6/0.6 ms
An important piece of information to note for now is that "0% packet loss" part. If you receive that, then you at least know that both your Linux box and router can talk together. If you don't receive something like that, then you'll need to check your Linux routing configuration, like this: netstat -nr. You should get something like this:
Kernel IP routing table
Destination Gateway Genmask Flags MSS Window irtt Iface
192.168.0.0 0.0.0.0 255.255.255.0 U 0 0 0 eth0
0.0.0.0 192.168.0.1 0.0.0.0 UG 0 0 0 eth0
If you don't receive something like that, then you'll need to reconfigure your network. There are lots of ways to do this, and I'm not going to get into all of that. (You can check out one of the Linux networking How-Tos for that.)

Nevertheless, I'll briefly show you how to reconfigure your network card with a static IP and using DHCP. For a static IP, do the following: ifconfig eth0 down; ifconfig eth0 192.168.0.2 netmask 255.255.255.0 up. Using DHCP do this: ifconfig eth0 down; pump -i eth0. In either case, you should be able to ping your router by now.

Next, you need to check to see which DNS server(s) your Linux box uses in order to resolve addresses. If you set up your network using DHCP, then you can probably skip this step. If otherwise, then edit /etc/resolv.conf (in your editor of choice). Make sure the DNS server entry (or entries) listed point to the same one(s) that the DNS entry (or entries) for your router.

Afterwards, you should be able to surf the web with Konqueror, Firefox, or another web browser.