The Vietnamese Ministry of Transport has created a master plan for Hanoi and Ho Chi Minh City (HCMC) which includes building a metro for the two major cities. The first lines shall be launched in 2016 (Hanoi) and 2017/18 (HCMC). Until their completion you must find different modes of transportation.
Although the traffic in the major cities seems to be very chaotic and inscrutable, buses are comparatively fast and also very cheap. For VND 5,000 - 10,000 (about €0.30 - €0.50) you can go from one place to another. They tend to be very crowded but nevertheless it is really quiet inside and most of the time (outside the rush hour) they are reliable. You have to take care of your valuables since it is an easy opportunity for pickpockets. Also make sure to get off at the stops quickly because the drivers hardly stop at all. In Hanoi the bus service is more frequent than in HCMC.
Taxis may be the easiest mode of transport and a great option if you heed some advice. First of all, you should consider only the major companies, although many more offer their services. It is best to arrange the price before starting because the taxi meters sometimes run faster for foreigners. It also protects you from “special routes” the taxi drivers may use in order to get more money.
It appears that the motorbike has displaced the bicycle as most used vehicle. Hanoi has over 2 million of them registered. You can either rent one or you hop on one of the motorcycle taxis (xe om). For the former you have to have very good skills and at least some experience with the local traffic. Otherwise you will soon get into trouble since the law of the jungle seems to reign on Vietnamese streets.
The xe om is an exciting alternative but similar to taking normal taxis you should arrange a fixed price beforehand. Insist on the price later since they sometimes try to negotiate it again. Make sure they have a proper helmet for you and as far as you can judge, satisfy yourself the driver is not drunk because they tend to have beer while waiting for customers. Besides these concerns it can be a great way to have a personal tour around the city.
You can take the cyclo-pedals, a rickshaw pulled by a bike, if you just plan on getting around the city. It is slower but you actually have time to watch some of the sights during the ride. Again: make sure the price is set before you start. VND 50,000 (~ €2) is a reasonable but realistic price for a one hour ride around the city.
Be aware that you always have to keep your eyes open when riding a bicycle or walking around the city as it often appears that cars or buses have right of way whenever they want to take it while smaller vehicles are often ignored.