Compaq Presario R3000 stays in sleep mode

The problem is probably that you're using faulty graphics drivers. Try booting into safe mode, uninstalling all graphics drivers, re-booting back into safe mode, using a program called Driver Fusion to remove anything from the graphics drivers that the uninstalls missed, and then re-booting into safe mode and repeating as necessary. All AMD, Intel, and Nvidia graphics drivers should be completely removed.

Once this is done, try installing the newest graphics drivers for your system. Don't bother using HP's website, go strait to the AMD website and install the newest driver that is compatible with your graphics card and your operating system. This is fairly strait-forward if you use the driver finding drop-down menu at the upper right of AMD's website:
http://www.amd.com/us/Pages/AMDHomePage.aspx

You could use the auto-detect and install tool from that drop-down menu if you don't know what graphics model your CP R3000 has.

If problems still persist after that, then report back and I'll see if I can make any other recommendations.