Hey ricky. Wow, you say "your job is causing you to leave the U.S.A." as if it were a bad thing. You have no idea of how many Americans would kill to be in your shoes right now my friend.
As for apartment hunting, there are a few good resources. Rent prices depend on whether you want to live alone or with other people.
A room in a shared flat can be about 40.000-50.000 ptas./month (approx. $200-$250) while a decent 1 bedroom or studio on your own in a nice area will go for between 70 and 90.000 ptas./month.
Keep in mind that apt. hunting these days is not easy in Madrid duer to the fact that many owners are selling theri places rather than renting them because with all of the "black money" (illegall earned dirty money) floating around that has to be changed into euros soon, they can get much more than their apartments are worth if they sell to someone who needs to invest the dirty money rahter than having to declare it when they change it into euros.
Anyway, the best time to find a place is in July and August for that is when there is a huge student exodusa from Madrid and many apts. become available and owners get desperate.
Try the following for a list of apt. hunting resources: www.multimadrid.com/mm_interactive/links/apts.htm
It has some pretty good links to In-Madrid, Segundamano, Anuntis, RoomMadrid, etc...