Vietnam is a relatively safe country to visit. As a woman, I have traveled extensively in Vietnam on my own with absolutely no problems. Despite people's fears, there is almost no animosity towards Americans.
That said, there are increasing instances of theft, especially in Ho Chi Minh City where pickpockets and snatch thieves on motorbikes are the worst menace. The best tip is to be vigilant at all times. Often cute kids or old grannies have deft fingers. Leave all valuables (expensive watches, jewellery, glasses, etc.) at home, and don't even wear flash costume jewellery. Make sure you have a firm grip on cameras and shoulder bags at all times and never leave anything you value lying around unattended. I would also not advise taking cyclos late at night, especially in Ho Chi Minh City or as a female on her own.
For everyday expenses, I recommend carrying a mix of US$ cash and dong. For larger items (hotel bills, train tickets, etc.) or when the exchange rate works in your favor, use dollars. For cyclos, local food stalls and small purchases, it's best to use dong. In either case, make sure you always have a stock of small notes so that you don't have to worry about change.
Almost everything is negotiable in Vietnam (with the notable exception of meals) and bargaining is very much part of the Vietnamese way of life. All tourists are regarded as wealthy - which we are compared to most locals - but that doesn't mean you'll always be quoted an outrageous price; small shopkeepers and restaurateurs will often charge you the local rate.
Finally, ask the receptionist at the hotel you stay about restaurant, bank, where to go shopping. It's reliable.
