A number of people are having an issue where upon upgrading to the latest version of iTunes they are not able to connect to the iTunes store. The program will claim there are problems with your internet connection and will state something like "iTunes could not connect to the iTunes store. Make sure your network connection is active and try again."
In addition to the error message, you may also get an error stating "We could not complete your iTunes Store request. An unknown error occured (112222)." when attempting to connect. This problem seems to be affecting only Macs running the latest version of iTunes, since PCs, iPads, and iPhones on the same network will connect just fine.
In a rather lengthy Apple discussion thread on the issue, the one thing in common most affected people have is they are located in the UK, Ireland, or Scotland, and are using "TalkTalk" as their ISP, indicating this could be an issue with Apple's servers in the UK or an ISP-specific issue.
Here are some options people can try that have helped some so far (after each suggestion try rebooting into Safe Mode, restarting the system again normally, and then launching iTunes again):
Reset iTunes' warnings and caches
Go to the "Advanced" section of iTunes' preferences and click the buttons corresponding to resetting warnings and the iTunes cache. This seems to be the most successful fix so far.
Clear iTunes preferences
This may be an issue with corruption in the iTunes preference file, which is located in the /username/Library/Preferences/ folder and is called "com.apple.iTunes.plist." Quit iTunes, remove this file, and relaunch the program and try connecting to the iTunes store again.
Clear user caches
Use a maintenance utility like OnyX, MacKeeper, Yasu, or Leopard Cache Cleaner to run a full cache cleaning routine on the system (optionally do a full "General Maintenance" routine).
Reset network connections
Try resetting your network setup by creating a new location in the Network system preferences and only adding the network port that you use to the list of network services. Then configure that port for your router/ISP and try connecting again.
Turn off Genius
A couple of people have had success by turning off the Genius feature in iTunes, which can be done in the "Store" menu. Additionally, try turning off other features like Home sharing if you have it enabled. If you then are able to connect to the store you can turn Genius back on.