Any ideas to spend a week besides standard tourist things?
The single best non-touristy thing I can recommend is to take a heritage tour with Tim Doling. His website: http://www.historicvietnam.com/saigon-and-cholon-heritage-tours/
I've met him when I was in Ho Chi Minh City, and I've never met anyone with such a breadth of knowledge for the city. He's fluent in Vietnamese, and has done a staggering amount of research about all the buildings, districts, and old haunts. As the city changes -- fast, so fast -- he's a great person to spend an afternoon with, since he can explain how it was during colonial times, and where it's heading now.
You won't see the city the same way after the tour, I promise.
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Ho Chi Minh City city is full of artistic coffee shops! Thousands! From a coffee shop deep in a maze of Vietnamese's road, to coffee shop that is build in an upper floor of an abandon place. You should definitely visit those coffee shop and try the traditional Vietnamese coffee.
Explore the Cu Chi Tunnels. It was a really interesting experience for me.