wireless internet connection established computer not connecting to internet

E-mails are showing up through MacMail. When I try to load a page, the blue loading bar loads about 1/16th of the way, and then just stops, and the little spinning circle (not the rainbow one, the one that shows a page is trying to load) just spins and spins and spins. Eventually I'll get an error message that the server where the page is located isn't responding, but the pages load on other computers, most likely indicating it's not a problem with the server?

I just tried doing a software update, but get an error message that Software Update can't connect to the update server...

Thank you so much, Linc.
I trust that this will make more sense to you than it does to me?

What is this console thing? A log of all the activity that happens when your computer tries to make an action?

1/21/13 2:47:24.000 PM kernel: AirPort: Link Down on en1. Reason 8 (Disassociated because station leaving).

1/21/13 2:47:24.000 PM kernel: 7881.814085: setDISASSOCIATE ****STA SYNC DISASSOC SUCCESS

1/21/13 2:47:24.000 PM kernel: The USB device BRCM2046 Hub (Port 1 of Hub at 0xfa100000) may have caused a wake by issuing a remote wakeup (3)

1/21/13 2:47:24.000 PM kernel: TBT W (1): 0 [x]

1/21/13 2:47:24.000 PM kernel: The USB device Bluetooth USB Host Controller (Port 1 of Hub at 0xfa110000) may have caused a wake by issuing a remote wakeup (3)

1/21/13 2:47:24.000 PM kernel: 7881.979841: performCountryCodeOperation: Not connected, scan in progress[0]

1/21/13 2:47:24.000 PM kernel: HID tickle 203 ms

1/21/13 2:47:25.144 PM UserEventAgent: CaptiveNetworkSupport:CaptivePublishState:1211 en1 - PreProbe

1/21/13 2:47:25.145 PM configd: setting hostname to "Emilys-iMac.local"

1/21/13 2:47:25.148 PM configd: network configuration changed.

1/21/13 2:47:25.000 PM kernel: en1: BSSID changed to 00:18:01:a9:6a:a3

1/21/13 2:47:25.000 PM kernel: AirPort: Link Up on en1

1/21/13 2:47:25.000 PM kernel: en1: BSSID changed to 00:18:01:a9:6a:a3

1/21/13 2:47:25.683 PM hidd: MultitouchHID: device bootloaded

1/21/13 2:47:28.270 PM configd: network configuration changed.

1/21/13 2:47:28.274 PM configd: setting hostname to "Emilys-iMac"

1/21/13 2:47:28.299 PM UserEventAgent: CaptiveNetworkSupport:CaptivePublishState:1211 en1 - Probe

1/21/13 2:47:28.300 PM UserEventAgent: CaptiveNetworkSupport:CaptiveStartDetect:2343 Bypassing probe on Robert1 because it is protected and not on the exception list

1/21/13 2:47:28.300 PM UserEventAgent: CaptiveNetworkSupport:CaptivePublishState:1211 en1 - Unknown

1/21/13 2:47:28.302 PM configd: network configuration changed.

1/21/13 2:47:28.000 PM kernel: en1: BSSID changed to 00:18:01:a9:6a:a3

1/21/13 2:47:28.661 PM airportd: _doAutoJoin: Already associated to “Robert1”. Bailing on auto-join.

1/21/13 2:47:28.927 PM configd: network configuration changed.

1/21/13 2:47:54.285 PM com.apple.backupd: Starting standard backup

1/21/13 2:47:54.287 PM com.apple.backupd: Error -35 while resolving alias to backup target

1/21/13 2:48:04.299 PM com.apple.backupd: Backup failed with error: 19

1/21/13 2:50:15.761 PM com.apple.SecurityServer: Session 100008 created

1/21/13 2:50:20.065 PM sandboxd: ([334]) com.apple.qtkits(334) deny file-read-data /Users/happilyevolvingsoul/Movies/iMovie Events.localized/Me Doing WoodWork/iMovie Thumbnails/MeSplittingWood

Let's hope this secret code tells some dirty little secrets about the problems with my computer!!
Thank you, again!

PHY Mode: 802.11g
BSSID: 00:18:01:a9:6a:a3
Channel: 11 (2.4 GHz)
Security: WEP
RSSI: -69
Transmit Rate: 48

Now you're getting tricky on me!
It appears that I need to know my routers username and password... The router is many many years old. I'll see what digging I can do, but chances are I'll need a new router.

Until I can contact Verizon, I'm at a standstill... but I deeply appreciate your assistance! You have been most helpful. Thank you.