Will any Cat5 Ethernet cables work equally well for Gigabit ethernet ports?

You need to have CAT5"e" or better yet CAT6 cabling for best performance on the LAN to support Gigabit ethernet.

+Currently, I use a Cat 5 100Mbs cable to connect the modem to the AEBS. So, I'm wondering if I can get faster throughput with a Cat 5 1000Mbs cable (if such a thing exists).+

Won't make any difference unless your modem is capable of speeds faster than 100Mbs. If you are replacing other cabling at the same time, then it would make sense to go ahead and replace the cable from the modem to the router as well.

+Are Cat5"e" or Cat6 cables readily available from major electronic stores, such as Best Buy?+

Sure.

In addition to Bob's comments, it's important to understand that Gigabit Ethernet requires using all four pairs of wires inside the cable ... regardless of whether that cable is CAT-5, CAT-5e, or CAT-6 certified. The real difference of the categories is how high of a frequency can they support. For example CAT-6 is rated for 10 Gigabit traffic. It's not that CAT-5 won't work, but it's not guaranteed to do so.

If you have any Ethernet runs in home that is supplying connections for both data & phone lines on the same cable, you will NOT achieve Gigabit speeds on those cables as the installer used at least one of the necessary pairs for the phone line.