I'll give you a more specific description of what I have done thus far:
I first bought AirPort Express, and when I first plugged it in-- after following the specified procedures--my laptop recognized it via AirPort Utility. I followed all the procedures listed. I also clicked on the option for getting it to work as an addition to the existing network.
I continued the setup process and thought things were going swimmingly until a status bar popped up showing that AirPort Utility was checking for existing AirPort devices (please remember that AirPort Utility [APU] had earlier recognized that the device was there, thus allowing me to start to setup process in the first place). It then said that no devices were found and, as a result, the whole setup process had to be aborted. Additionally, I lost all wireless connection that I had previously, and I had to call Verizon to have that reestablished.
When I tried to repeat the setup procedure, APU continually stated, right from the beginning this time, that no AirPort devices were found. I called Apple Support and a woman gave me instructions that enabled me to wirelessly transmit to AirPort Express via hard wire from the modem. However, when the ethernet cable was removed, the wireless ability failed. She then stated that she believed that I would be able to get things to work by using AirPort Extreme, which uses ethernet to connect it to the router/modem.
I then bought AirPort Extreme and subsequently had the exact same setup problems that I listed in paragraphs two and three of above. In other words, things went OK until the end when APU stopped recognizing Extreme. I also lost my original wireless ability again and had to call Verizon again to have that reestablished.
I then tried to set up AirPort Extreme again (this time after resetting Extreme-- which was the only way I could get APU to recognize it from the beginning), using the option to add a network. The same problems occurred in that APU eventually stopped recognizing the Extreme toward the end of the setup process; however, I did not lose my original wireless connection that I had lost previously during these types of procedures.
I have not attempted to get Express to work with Extreme, because I can't get Extreme to work.
Summing this up, I would say that until I find out why APU eventually fails to recognize both of these devices, I'm not going to be able to do much.
Thanks for the links and your input--any additional ideas are welcome.