How can I pass cables vertically through walls that contain fire blocking?
There are a variety of methods of fishing wires through walls.  The most crude being brute force with some cable types.  Coax can often easily be pushed down walls.

The most common way to fish cable is a fish tape:


(source: homedepot.com)  

This is a flexible steel or fiberglass line that is wound up.  You extract it by holding the reel and pushing the handle.  To wrap it back up, you do the opposite motion.

In some cases (most often in drop ceilings) you can use rods to pass the cable around:


(source: homedepot.com)  

If it is a short distance, the cheapest and easiest thing to use is a metal coat hanger. 

With all methods, you feed it from one area to another and then pull the wire back out.  It doesn't usually matter what direction you go, but you will find that with some routes, approaching it from one direction is easier than the other.  If the route doesn't match up with how you are feeding the wire (ie: it is coming from one side, but for whatever reason you need to fish from the other), you can pull a temporary line (nylon string) through with the fish tape, and then pull the wire back through with the temporary line.

Plan your routes well using a stud finder and other visual hints.  If you are lucky you will only need two holes.  In some tough scenarios you might need to cut another hole to allow you to pull the wire through.

A flexible drill bit can also be useful:


(source: homedepot.com)