Hi Ric:
I've seen several problems with this HP printer thing, including on my home setup. First thing that comes to mind is how are you connected your HP desktop to the printer? Method #1-USB; Method #2-Ethernet hardwire cable; Method #3-Wireless. You didn't mention that. I've used the 309 series and I currently have it's replacement, the 310n here on my Dell desktop. If you are connecting to your 309g via your wifi router, or if you are connecting directly to the 309g via wireless peer-to-peer you could also be having networking problems with the 309g.
I suggest you temporarily reinstall your 309g software using either Method #1 or Method #2; both use a physical cable. If the problem goes away, you'll need to repair or replace the 309g. If you experience the same problem with Methods #1 & #2, the problem is probably in your windows networking components. If you cannot get it to work via System Restore, you may have to completely reinstall your Windows7 from either the HP Recovery Partition hidden your hard drive or from Factory Recovery Discs you would have had to make since you unboxed the computer new--if you haven't you should make them NOW, as HP only gives you ability to make 1 set. Ensure you use quality CD or DVD media; you don't want to scrimp on discs when trying to rebuild your system.
If you find that your HP desktop is the culprit and a Windows recovery is in order, ensure you backup all your critical data to either external hard drive, flash drive(s), or CD/DVD media PRIOR to your rebuild. Upon completion of your rebuild attempt to install your 309g software on both Admin and Family accounts *yes, you still have to do that* and see if problem is still there or has been resolved. If so, your Win7 was scrambled, and you fixed it.

Proceed to re-hook up your wireless setup with 309g if that's how you were running it originally; and try again.
Once you fix the problem and are connected to your 309g via Method #1 or #2 and all accounts can properly access the printer, getting the wireless part working should be fairly straightforward if you're pretty savvy with networking. You should be set to go at that point!
One thing to remember, if you're after reliability, home wireless printing is still pretty iffy, and you'll get interrupts and disconnects randomly much more frequently than using a hard line cable such as USB or ethernet. So unless you are printing to a downstairs printer from an upstairs location like a bedroom or den, you may consider putting in a hard line cable to keep things from mucking up on you. The 309 and 310 series randomly lose wifi connections as do most of the HP wifi printers. I see this all the time in businesses as well as people's homes. Just some thoughts for you to consider after you get it fixed and running again. Remember every time your ISP takes a hit due to an outtage on their part, or you have a power failure on your cable modem or dsl modem, they often lose their wifi settings and you have to reset BOTH your wifi router and the wireless settings in your printer *pain*. So keep in mind my suggestions.
Let us know how it turns out.
Luck!
BIGBEARJEDI