Generally speaking, and as is often the case on the whole of the Asian continent, Vietnam is a safe country which does not pose any particular safety problems. As with everywhere in the world, there are dishonest people but with a small amount of caution your trip to Vietnam will proceed with no hiccups.
There is no reason why your trip to Vietnam should go bad or that you should be at all worried. However, the main problem you could face would be to be the target of a thief or a pickpocket. This is especially possible for public transport during your journeys or whilst walking in certain parts of Hô Chi Minh city.
The mass arrival of tourists has also brought about the appearance of a petty crime. However, cases of violence are absolutely an exception. You do not have anything to fear in terms of this.
Do be careful though about people trying to rip you off or scam you, which may happen more regularly. A shopkeeper might try to make you pay a little more, and a taxi driver might do the same, taking a different route to what was planned when you booked it with the local agency.
Even though Vietnam is completely safe, situations can easily turn bad if you step foot into the world of drugs. Then you risk crossing paths with true criminals, a lot more dangerous than the petty delinquents in Hô Chi Minh city.
All you need to do for a danger-free trip is to be a bit cautious. On public transport, keep an eye on your bags which you should keep locked shut.
In the streets, all you need to do is not dress like a typical tourist. No camera around your neck and no ostentatious signs of wealth. Nothing that can make you the ideal target.
In the rare case of violence, don't try and act like a hero. Stay as calm as possible and co-operate without shying away. Always make sure not to put all of your money in the same place. Have multiple hiding places and keep a certain amount easily accessible which your assailant could be happy with.
To not pay too much, try to inform yourself about prices in advance and always negotiate. Same with taxis: ask the taxi driver to put the meter on, or agree on a price for the journey before leaving.
As for drugs, you are old enough to know the dangers and consequences for yourselves. Simply don't go near this environment.