Generally BSOD don't occur unless you are using shoddy unsigned drivers, or there is a hardware problem.
If you are not using unsigned drivers, then I suggest stress testing your PC. Memtest will do a good job of testing your memory, and Prime95 will put your CPU through its paces.
You can always try reseating your hardware by removing it and blowing out the connectors to make sure there is no dust.
This pc is basically brand new--only been out of the box for a month...and my problem is I know very little about computers--so should I just get on the phone with HP Tech Support? If not, are Memtest and Prime95 free downloads and can a guy with very limit computer skills handle the job at hand?
Also, I found this while doing an internet search...I guess this wouldn't fix the problem: