HP printers are well supported in Linux. Since this thread is in the Mint forum, just open the software center or Synaptic and look for anything with "hplip", especially the GUI part. Install the packages, setup the printer, start printing.
O.K., you may need a little more hand-holding, but in general, that is the approach. Start there, and when you see something that doesn't look right, come back with questions.
Also, I'm not sure where to find the software center or Synaptic. I'm BRAND NEW to the Linux world. So step-by-step instructions on not just what to do, but how to get there (what to click, etc.) would be very helpful. And thank you all so much for taking the time to assist. I really appreciate it.
This may be due to existing firewall settings blocking the required ports. When you are in a trusted network environment, you may open the ports for network services like mdns and slp in the firewall. For detailed steps follow the link. http://hplipopensource.com/node/374