Dear Harlan, For many important languages, e.g. English, Spanish, French, you name it, we have been brought up into our mother language by our unchosen teachers at home and we all think we speak a neutral form of it. There are more than 20 Spanish speaking countries, including the U.S. Usually, we are not too exposed to the variants spoke in other countries except for some movies and celebrities, and it's mainly focused on pronunciation. In any case, we all have some small difficulties at first catching up with some usual words and constructions that are different from our home country. That is normal, not so cumbersome and fun to experience. Try though to learn the standard form of the language, avoid slang, because that is too local and stick to one form of it. Enjoy your learning.

Yes, just make sure -especially if you are a beginner- that your teacher doesn't speak too informally or with a strong accent, rather choose a more neutral one, and don't use too much slang, as this is completely different in each country of Latin America. Saludos!

I hope the teacher is responsible enough to teach you a general Spanish and to avoid problematic vocabulary that may vary from one country to another. I'm sure that every Mexican will understand you!

of course, but remember each country has its own slangs

You will be able to talk to anybody that speaks Spanish. In the Spanish speaking world, there are only slight differences in pronunciation and almost no differences in grammar. The only significant differences are in the vocabulary used, specifically the words for food, clothing, everyday items, and idioms.