Welcome jt_glass to wattOS.
The HP 2710p is well within required min spec, so that ought not to be the issue.
Black screens and Ubu 14.04 are hardly unknown, and sometimes unraveling the mystery can be difficult, even for the Ubu techs.
Let's take a guess that it's browser-caused and try for a solution :
Open Chrome. Type chrome://settings in Chrome's address bar and hit enter.
Then go to the bottom of that page and click on Show Advanced Settings.
Then go to nearly the bottom of the new page and see if "Use hardware acceleration when available" is ticked. If it is, try the Web Store after you untick that setting and restart Chrome.
If that resolves the issue, then thanks go to the people at
http://ubuntuforums.org/showthread.php?t=2247572 I think the corresponding action in Firefox starts with about:config in the address bar.
Other assorted notes.
Midori is brilliant for some users and useless (eg crashes) for others. I'm unaware of the reason. It might be unrelated to this issue.
For 1GB of RAM, you'll often see the 32-bit version recommended (notwithstanding that you're within spec).
If the above doesn't work, other actions would include ensuring your install is uptodate, running some music (eg internet radio) in the background to see if that too is cut when you use the browser, and checking whether ctrl-alt-delete or c-a-F1 bring some response.
Good luck. Let's hear how it goes.