First, to ID your version of Vista, go to Start / Computer / System Properties and it will tell you. Please either update your profile or post the answer here. It should be said something like "32-bit Vista Business SP2" or whatever is appropriate. It won't say it that way, but it will contain all the information for you to be able to say it that way. Also, please provide the make and precise and complete model number of the HP laptop - I may need this to research your computer and want to be sure I'm checking the correct one. This will help us.
In the meantime, let's see if we can figure out your internet problem. Please go to Network Posting Tips
and follow the instructions. One part will result in a screenshot to post here and the other will be a batch file you will run to reset common causes of networking problems.
If the batch file doesn't fix the problem, perhaps information from the screenshots will help me identify it. Has it ever worked or is this the first time you've tried? What internet browser are you using?
Since there is an indication that there may be an issue with your network adapter (could you please include the EXACT diagnostic report in your next post - word-for-word), I recommend you go to the computer manufacturer or the adapter manufacturer's website and obtain the most recent device driver, firmware, and/or software updates for that specific device and your version of Vista (which you discovered above). Go to Device Manager by going to start / search box and type device manager and enter and then double-click on the program icon that appears. Check each device for a red x, yellow ! or white ?. These identify devices with problems (probably drivers, but also conflicts or something else). Click on each for further details and troubleshooting tips. If you need to get drivers (and you should for the network adapter), do so from the computer vendor or device manufacturer (NOT from Microsoft Updates). Proceed as follows to get the drivers: http://pcsupport.about.com/od/driverssupport/ht/driverdlmfgr.htm. Once you have the drivers, you can install them through Device Manager as follows: http://www.vistax64.com/tutorials/193584-device-manager-install-driver.html. There are other things we can check and try, but for now I'd like to see these results (and those screenshots) and answers to all the questions I asked before proceeding any further. I suspect it is a simple setup problem, but want to be more sure before recommending anything further. I hope this helps. Please post back in any case, but if not the information you provide will help us identify the problem. Thanks and good luck! P.S. The following articles may be helpful in setting up the connection if it's merely a configuration issue. I recommend you read or at least scan them to verify you did everything right and to see if they provide any hints on options to try on your own in the meantime that may resolve the problem. Sometimes deleting the entire connection you have and starting fresh with a new one after reading the articles on how to do it and problems that can occur in doing so can do the trick - but that's entirely up to you. I'm not suggesting you go that far at this point (though you could create a NEW connection after reading the articles without touching the current one to see if that works). If so, you then need only remove the one that didn't work, use the one that works, and you're home free. Different devices use different setups. In some cases, connecting locally automatically connects you to the internet at the same time (as with mine now). In other cases, to connect to the internet you need a separate internet connection (as with me a month ago). This can usually be verified by your ISP based on the modem (which was changed and why the process changed) and router and setup. We'll get into that later if necessary. Here's a tutorial on how to setup a home network (including a wireless one) - check out the additional links for further information that may be of great help: http://windows.microsoft.com/en-US/windows-vista/Setting-up-a-home-network.
Here's an article on troubleshooting wireless network connections (again with helpful links): http://windows.microsoft.com/en-US/windows-vista/Troubleshoot-network-connection-problems.
Here's a long guide on wireless internet connections in Vista: http://www.techwarelabs.com/guides/misc_mod/vistaconnection/.
Here's an article on how to setup wireless internet connection in Vista: http://www.clpgh.org/usingthelibrary/technology/wireless/WindowsVista.pdf.