When you go to Safari > Reset Safari, you have the ability to deselect any items you don’t want to reset. You can find a full list of what those options are in the following document, which should mostly still apply to Mavericks (with a few exceptions) even though it says Mountain Lion:
Thanks for that. I now know where to start, but I'm still cautious about using that command. Since my problem is with websites that I can't open with Safari, I'm going to investigate some of those other links before deciding what to do. I have good reason for this timidity, since I just recovered the use of my computer after four days of extremely frustrating and time-consuming efforts to reconnect with the Internet. That ordeal started when I clicked on a button on my router at the instructions (or so I believed) of a Sony technician. That turned out to be the wrong button, although it was the only one. As a result, besides the four wasted days, I had to pay Linksys $29.99 to reset the router password (my service warranty had expired). Why did this take four days and at least ten technicians to discover and fix this error? Don't ask. I don't want to relieve that experience. I'm stoill recovering.
One other thing occurs to me: for YEARS I had trouble connecting with many sites. It never occurred to me that the problem might be with Safari, so I was forever cussing-out those sites for not doing what websites are supposed to do, or so I thought. Anyway, finally someone, I don't remember who, suggested that I repair Safari by trashing and reinstalling its "Preferrences". I did and was immediately able to access all of those websites that had been locked to me. This time that thought finally reoccurred to me, but I didn't do it because I thought that it might be possible to make a mistake in the process and do some real damage. I was going to wait a little longer before getting someone to walk-me-through the process. Anuyway, thanks again for your help. I say "help" because I'm not yet confident about using the RESET button. I did make some changes with the Safari control panel, and maybe that did the trick. We'll see.