I?ve been toying with the idea of learning Spanish. (I live in Tampa, Florida.)

I remember, though, the difficulty I had learning French way back in
high school.  The written grammar drove me nuts.

I wish there were a way to learn a language the way children do: 
simply by listening to their parents.  I don?t think I need to write
Spanish, at least not right now.  I?d just like to be able to engage
in limited conversation.

(1) Is there a respected language-teaching system that concentrates
primarily on speaking and hearing, and leaves the reading and writing
to a later time?

(2) Which version of Spanish should I learn if I wish to communicate
with people in Tampa (and perhaps the Caribbean)?

You might be interested in the Total Physical Response (TPR) approach
to language teaching.  Here's one website with more information on it:
http://www.tpr-world.com/

Much luck in your studies!

I am bilingual, I live in Mexico, I spent several years going to
english classes and after that I did not know anything. I spent a year
in Michigan, and after the first 3 months I have already learn it.

I see a lot of people here in Mexico waste a lot of time and money
trying to learn it in some school, it is only effective if you have a
big desire. If you can not go to a Spanish spoken country for a while
the best thing is this: movies or television in spanish with close
caption also in spanish, watch it until you understand what they are
saying, this is the most helpful because you see the exact situation
for a proper word or phrase.

A friend of mine learn english really well, watching american cable
with close caption.

Also, you may want to look into the Pimsleur series. The idea is that
you learn to listen and speak actively, and the series does not teach
any grammar (you are supposed to infer it from phrases and patterns
you learn.)
More information here: 
http://en.wikipedia.org/wiki/Pimsleur_language_learning_system