I think it's possible to learn the two (SG & HG) concurrently, depending on the circles you move in.

I moved to St Gallen to attend school, and while the classes were conducted mostly in High German, as soon as we left the classroom all exchanges were conducted in Dialekt. I made every effort to learn to communicate with my peers in their mothertongue, and it was always appreciated. Everyone I met was chuffed when I spoke SG, which made learning easier as more people were prepared to invest their time patiently helping me learn. I always find people are more likely to help you if they see you making an effort.

A few years after I moved back to Australia, I was staying in a YHA while on holidays. I heard some guys speaking in SG and called out to them. We chatted for a few minutes, after which one of the boys said "You're from [name of village in Rheintal] aren't you? I'd know that Dialekt anywhere!!" We all had a good laugh! All my efforts seemed worth it when a native speaker picked the village where I had learned to speak Dialekt!
Honestly folks, make the effort to learn both. Out of respect for your friends and colleagues, learn to converse with them in their mothertongue. I don't think your efforts will be in vain...