As Richard astutely recommended, one of the first steps is to determine if you are using the correct drivers for the printer and the version of Vista you have. I checked, and believe you need to be using the following:
HP Photosmart Full Feature Software and Drivers HP Photosmart C6350 All-in-One Printer - HP Customer Care (United States - English)=. This includes both drivers as well as related software - both of which are needed to get the printer to function properly. One nice feature on the page seems to be the ability to check your driver to see if it needs updating. It can't hurt to try that. Even if it says it's OK, it may still be wise to update anyway (as this will also update the software as well) in case there is some corruption in the version you have (not enough to show as a problem, but enough to cause one).
The articles he included may also be helpful - especially the one that addresses this specific problem - and are worth reviewing and following whatever instructions they recommend.
If that doesn't work, then try downloading the print diagnostic utility for that specific printer at:
HP Print Diagnostic Utility HP Photosmart C6350 All-in-One Printer - HP Customer Care (United States - English)= and run it to see if it identifies any problems with the printer or the setup or anything it is programmed to detect (and it most likely offers solutions to fix those problems - perhaps even automated). I don't know if it will find anything, but since troubleshooting is its purpose, it certainly can't hurt to give it a try. You indicated you connected it to your computer, so I'm assumng it isn't a network problem; however, if a network is involved, there are additional diagnostic utilities for that at:
Software & Driver Downloads HP Photosmart C6350 All-in-One Printer - HP Customer Care (United States - English).
If that doesn't work, let's sort of start over from scratch just to be sure. I'm assuming you're using a USB connection from your initial post. Here's information on how to setup the printer hardware - check to make sure you followed all the steps correctly:
Setting up the All-in-One (Hardware) HP Photosmart C6350 All-in-One Printer - HP Customer Care (United States - English). Be sure you removed the tape completely from the print head or the cartridges will not function (and could cause the problem you've described). Next is installing the software and it must be done in a
precise order (attaching the USB connector ONLY when instructed - if done with the connector in place the whole time, the installation may not work properly (I've stumbled over this problem myself, not thinking it made any difference, but when I did it as instructed it began to work fine)).
Installing the Printer Software in Windows Vista for a USB Cable Connection HP Photosmart C6350 All-in-One Printer - HP Customer Care (United States - English). You may have already done this. If so, uninstall the software and then re-install it - connecting the printer when instructed in the proper fashion.
Are you able to print an alignment page from the printer itself (this is not controlled by Vista or your connnection with the computer but by the printer itself? If not, then the problem is with the printer and not with the connection, Vista, or the setup. Recheck the hardware setup procedure. If the problem continues, contact HP Technical support:
Contact Support and Customer Service HP Photosmart C6350 All-in-One Printer - HP Customer Care (United States - English).
If the alignment page prints fine (which verifies the printer can print something other than a blank page), then when you've completed all the steps, go to Start / Control Panel /Printers and select your printer and right click on it and click Properties and then click Print Test Page. If you get nothing, then the printer and computer are not communicating. I would recommend uninstalling and re-intalling the printer to see if that resolves the problem. If not, go to Start / Control Panel / Device Manager and confirm that there are no issues with the printer drivers and that they indicate they are functioning normally. Then double-check the printer software to make sure it is configured properly to allow printing. If so, I recommend at this point you contact HP Technical Support for assistance:
Contact Support and Customer Service HP Photosmart C6350 All-in-One Printer - HP Customer Care (United States - English).
I hope this helps. If not, post back and we'll see if we can offer any other suggestions - but I suspect if you get to the step of contacting HP Technical Support, they will get you up and running. If not, you could also try posting in the HP forums for assistance from people who use the product and may have ideas we haven't considered:
Printer, All-in-One Install and setup - HP Support Forum.
Good luck!