One more Spain travel question?

I have a dilemna. Well, I'm finally going this year and I'm planning my trip now... for November. I would like to get everything squared away now, though. :)

I'm going to study there, but my course lasts for a month. Here's my dilemna... my course is in Barcelona, and while I'm studying I'll have nearly zero free time. I really would LOVE to see Andalucia becuase that is the region I've been researching my whole life. After I finish my course I would like to go there and visit. The thing that attracts me the most is the architecture and the beautiful things to look at; the cathedral at Jaén, the cathedral at Sevilla (the one that Columbus is supposedly buried in), La Alhambra.

The problem is that I can't afford to stay very long... a month at most.
I would like to know: I know from friends that even a lifetime isn't enough to get to know that country... but to really "get something out" of my trip, how long would you recommend staying? Tourists CAN stay up to three months, and I would but I can't afford to.

I have read, however, that people who are not entitled to a work permit do have an option. I have read that there are organizations that match up families with people who are certified to teach EFL, and in exchange for teachign EFL for free, they provide you room and board and sometimes food. I looked up said organizatons on the web but the site is no longer valid. Anyone know about this?

I would like to go ahead and get airfare asap so I don't miss out on deals.

Gracias mil si me pueden ayudar.
Additional Details
Yes, of course I have my degree.

I should clarify, what I was wondering was how much time I should stay as a tourist to get to know Andalucia.

Umm thanks though Xavier... I think :)