I am using an older (2006) 54Mbps AirPort Extreme Base Station ... Can I use an AirPort Express to extend the range of the WI-FI and how exactly is this done?
The short answer is yes; the longer answer is that the only way to "wirelessly extend" the range of your 802.11b/g-only Extreme is to create a Wireless Distribution System (WDS).
Not only is a WDS a tedious pain to set up, even for experienced users, its performance is likely to be disappointing. It may not even solve your dropped signals or slow throughput.
There are better ways to extend your wireless range. By far the best and least expensive alternative is to connect an additional base station to your existing Extreme with an Ethernet cable. If you don't want to do that, and you understandably decide against a WDS, then your remaining alternative is to buy a newer, 802.11n capable Extreme or Express. They can use Apple's "extend a wireless network" feature that will eliminate nearly all the disadvantages of a WDS.
Read this to determine how to create a WDS: http://support.apple.com/kb/HT4262