In major cities and tourist areas — including Hanoi, Ho Chi Minh City, Da Nang, Hoi An, Phu Quoc, Sa Pa, and Ha Long Bay — you can wear basically whatever you want in public as long as you're not nude
In these metropolitan areas, people have been exposed to many Western clothing style through movies and tourists, and we are fairly used to see men wearing flip flops and tank tops, and women wearing very short skirt. Don't worry you won't be offending anyone if you wear very short garments
Just don't walk around topless, anything else will be fine
When you go to upscale restaurants, bars, and other private establishments, you will be expected to wear something more modest. Smart casual is fine, meaning jeans and shirt (or T-Shirt is fine too). Basically wear to these places what you would wear to same kinds of places back home
However, when you visit certain sites that require respect, including all religious and government sites (like the Ho Chi Minh Mausoleum in Hanoi for example), you will be expected to:
Cover your shoulders
Cover your knees
Cover your toes
No hats
A small number places require more formal wear, like the opera house for example. You generally won't ever have to wear formal as a tourist in Vietnam, but just keep in mind that any kind of places back home that require formal wear, you will likely have to wear the same thing in Vietnam as well
If for whatever reason you're going off the beaten path to visit the country side, you will want to dress more modestly if you're a woman. Avoid things that are too revealing, like bikinis. For men, there's not much constraint even in the countryside on what's acceptable to wear.
A bigger consideration for what to wear is to dress comfortably
Vietnam is a hot and humid country in most places you're likely to visit. So wear something light and breathable