of the cities you mentioned, Apex is "generally" the least expensive. there are several great little apartment communities that are reasonably priced, and nearly all if not all accept dogs and cats.
the price of apartments in Apex is highest at the top of 55 (Morrisville end) and the cheapest at the bottom (heading towards Fuquay, which is dirt-cheap comparatively, but a bit of a drive for you)
if you really want Cary, there is a complex that is safe and clean, but old and a bit run down, that friends of ours with money issues stayed in because the price was very cheap, and the location was great. I don't remember the name, but it's just behind the Whole Foods in Cary, you can probably find it online with a little googling