It's taken me a few days, multiple calls, and a couple trips to Best Buy, but thanks to the above suggestions, I can now print to my printer. Here's what it took in case others run into the same situation.
1. I did need to download the software. Do it as explained above.
2. I called HP which was a waste of 45 minutes. Don't bother doing that. You'll end up frustrated. They finally suggested that I purchase a new printer (however I knew the printer worked fine).
3. Follow the steps listed for the obtaining IP address. Get a print out of that as you'll need it.
4. It did require a different cable than before, which was suggested above. Note that I first purchased a serial to USB cable. Not only did that connection not work, the cord was only 5 inches long (which the sales person didn't tell me and I didn't catch that on the box). This meant that I had to hold my laptop at the back of the printer while attempting to print--the cord was too short to even put the laptop on the table with the printer.
5. Went back to the store for a parallel to USB cable.
6. Meanwhile, called Apple tech support. They were awesome. They walked me through it, using the information on the printout. They had me hook up the parallel to USB cable and the printer instantly printed.
7. About the cables. If you want to wait a few days, order it on Amazon.com, which listed the same cable for more than $20 less than Best Buy, and it has free shipping from Amazon.
Thanks to everyone for helping me figure out what to do to be able to print!