A couple thoughts:
Onboard graphics doesn't mean you can't upgrade to a better graphics card later. In fact, most motherboards now will allow you to use the onboard AND a PCIe card simultaneously, for multiple displays.
That's true, I can always throw in a dedicated video card down the road if needed or just transfer the one from the old computer into the new one for the time being
Simply pluggin in your old drive is not likely to work right off the hop. If the old machine was working, there's a tweak you could do that would allow Windows to work on the new chipset; otherwise you need to AT LEAST do a repair install of Windows. Ideally, Windows should be reinstalled clean now and then anyway, so while you're at it, get a new drive with the new system, install your OS and software on that, and just plug in the old drive as a secondary, so you can still access your old files. In fact, if you can afford the extra money, get an SSD for your new system drive.
This is what I was afraid of, hmm so you would install a new version of windows (7 or 8) onto the SSD and then I would be able to just plug in the old HD as a secondary drive and be able to access all my old documents and pictures and stuff?
Something like this should do the trick:
Samsung 840 Series 120GB 2.5in SATA3 MDX Solid State Disk Flash Drive SSD
lilaznviper is correct, your HP's power supply is probably shit - it's likely fairly low power, and the fans in them often fail over time. There's a good chance you'd have to replace the PSU before long anyway, so just do it now and be done with it.
Yeah I decided to get the case and PSU bundle
NCIX PC Case and PSU Bundle - Zalman Z9PLUS Gaming ATX Case and XFX 650W Core Edition PSU
The only things you should really consider transferring over are your drives (HDD and optical), and even the optical... well, those often fail over time as well, and a new Lightscribe DVD-RW can be had for under $30.
I don't really play DVD's or Cd's on here, I might burn a cd from time to time but that's about it