I usually start troubleshooting network problems by reviewing my network configuration. First, make sure that you are really connected to the network regularly, whatever your network app tells you, it might not be correct. You did not give details about your config, so i'll have to guess a bit. I guess that you get your internal IP (the network address in your home network) automatically from the wireless router (it's called DHCP). I also guess that the other computers that are connected to the network also use wireless (otherwise, if they are on cable, it would be a separate network, really). I also don't know Linus Linux Lite, so i can't give much GUI help. My first guess for your problem is, that your machine does not use the right gateway or DNS servers (look farther down). Ah yes, an expert i am not!
First, make sure that you are indeed connected to your own network, not to one of your neighbours'. Go into the config of your router (usually they can be configured with a web browser, a look into the manual will help). Give your network a name to distinguish it from other "linksyses" around you. You should also use WPA encryption or allow connections only from selected MAC addresses if you have not already done so (MAC is the unique hardware address of a network interface).
Open a shell, type in the command "ifconfig". Your output should look like this:
wlan0 Link encap:Ethernet HWaddr (your MAC address is here)
inet addr:192.168.0.5 Bcast:192.168.0.255 Mask:255.255.255.0
The important part is, that there shoud be an "inet addr" (your internal IP), not "0.0.0.0". It might happen that the network connection is not usable by all users (at least Ubuntu allows this setting). Try looking into your network connections applet if it has some checkbox, like "available to all users".
If you have an IP, and the network is available, try to ping the router. You have to know the IP of the router for this. In most cases it's the first address in the network. Mine is "220.127.116.11". Ping sends small packets to the given address and waits for a response.
$ ping -c 5 192.168.0.1
PING 192.168.0.1 (192.168.0.1) 56(84) bytes of data.
64 bytes from 192.168.0.1: icmp_seq=1 ttl=64 time=0.674 ms
64 bytes from 192.168.0.1: icmp_seq=2 ttl=64 time=0.641 ms
64 bytes from 192.168.0.1: icmp_seq=3 ttl=64 time=0.611 ms
64 bytes from 192.168.0.1: icmp_seq=4 ttl=64 time=0.656 ms
64 bytes from 192.168.0.1: icmp_seq=5 ttl=64 time=0.712 ms
--- 192.168.0.1 ping statistics ---
5 packets transmitted, 5 received, 0% packet loss, time 4001ms
rtt min/avg/max/mdev = 0.611/0.658/0.712/0.046 ms
You could do more pings, you should not have any packet loss and a very short round trip time (rtt). If there is packet loss, what is your signal strength? Move closer to the router, then the signal should be better. Move the router antenna into a vertical position. If the router has two antennae, put both at different angles.
Next, if you can ping the router, you should also be able to ping one of the other computers in the network. If that also works, you should be able to ping a computer outside of your network.
$ ping -c 1 ecosia.org
PING ecosia.org (18.104.22.168) 56(84) bytes of data.
64 bytes from server1.eco-server.net (22.214.171.124): icmp_seq=1 ttl=54 time=36.6 ms
--- ecosia.org ping statistics ---
1 packets transmitted, 1 received, 0% packet loss, time 0ms
rtt min/avg/max/mdev = 36.676/36.676/36.676/0.000 ms
In your case, if you have the problem i guessed first, this last ping probably says "Ping: unknown host ecosia.org". Try looking up the domain name ecosia.org:
$ nslookup ecosia.org
If your DNS (domain name service) is not set up correctly it might say "name service not available" or similar. Typing the address "126.96.36.199" into the firefox address bar should bring up the google website, anyway (note: "188.8.131.52" fails to bring the ecosia homepage for a reason i don't know). If it doesn't, then in your network settings, the gateway is not set correctly. Try to make a manual configuration, type in a fixed IP for your machine (it should match your internal network, so probably "192.168.xxx.yyy" or "10.0.0.xxx", again to get from the router settings, numbers going from 1 to 254). As subnet mask, type in "255.255.255.0". Gateway yould be the IP of your router (which in most cases has the number "xxx.yyy.zzz.1"). As DNS server, try also the address of the router. Sometimes that doesn't work, try to type in the DNS server(s) of your internet provider directly (should be availabe on the providers' website).
With the manual setup, the connection should finally work. You can now configure your wlan router to correctly give gateway and DNS addresses with DHCP, or make correct DNS forwarding to your providers' DNS.
HTH a bit.