I downloaded what I believed to be the basic installer and tried to install Limewire on a Mac OS 10.5.2. When the installer opens, the button to "continue" at the very beginning is not highlighted and can't be clicked on. Does anyone know how to correct this?

You need to be logged into Admin account on mac OSX to be able to install programs.
Normally if several HDD's or partitions, the installer may take some time to check install permissions on each so it may take a little time before it shows options. Once they show (not greyed out), then you click on disk of your choice to install to, then click Continue button.