Wow, all this response is great! Although I'm still no closer to solving my dilemma. I'll give more information.

Basically, my girlfriend (whom I may ask to marry me) is Swiss and we're going to meet the parents and I guess I'll be spending time with her friends there too. (this is in Zurich by the way) So yes, I realize I'd be fine with just English, but I want to learn the language as a cultural gesture more than anything else. I assume that Swiss German is the best way to achieve this?

But as DM pointed out, High German seems much more useful. So would speaking High German earn me any cultural points with the Swiss, or is it no better than just speaking English?

Are they really that different? If I learnt one of them, then would it be easy to pick up the other one?

I'll be coming there in about 3-4 months.