Tram: The easiest and, therefore, the most common way of travelling around the city is by tram. There are, in fact, two prominent tram lines running between the Central Business District of Hanoi and The Old Quarter: Tram Line 2 and Tram Line 6. Both lines run through the centre of the city and stop at all major parkland at famous sites, including Hoan Kiem Lake, Temple of Literature, and other historical places. You can buy tickets either at any tram station or at any ticket office in Hanoi.
Bus: Travel by bus from anywhere to between two or three locations in the Old Quarter covering destinations such as Hoan Kiem Lake, Hanoi University, and An Giang Street. One can either check schedules, fares and route maps online or with apps like "Vietbook" or "Tieto."
By taxi: If you would prefer a more luxurious and private means of transport, the taxi offers that. The approximate cost of a taxi ride from the airport to the Old Quarter is around VND 80-150 ($0.40-$1.00 USD). However, please bear in mind that there can be heavy traffic during rush hours at weekends and on holidays traffic-tidings manifold, and do better if an advance booking can be organized.
1. Taxi:
The taxis are outside the airport terminal. Use reliable taxi companies such as Mai Linh or Vinasun. Expect to pay a fare of around Vietnamese Dong 350,000-400,000 (~15-17 Dollars). The journey could take anything from 45 minutes to an hour to most of the destination in the city center. Price depends highly upon traffic conditions.
2. Airport Shuttle Bus:
The airport shuttle bus (Bus No. 86) has great frequency and a relatively low price. The fares for this bus are about 30,000 VND (that's around $1.30 USD), and it takes 1 hour to get to the Old Quarter. The bus will halt at many stops along the Old Quarter, so keep an eye out to find which stop is nearest to yourselves.
Take Bus Number 18 from the airport to My Dinh Bus Station, and then change to Bus Number 10 or 17 to go to the Old Quarter. This is the cheaper alternative but not so convenient because it involves so many transfers, which may be a real hassle with lots of luggage or no command of Vietnamese. The cost is about 20,000 VND one-way, but it could end up taking about 2 hours because of stops and other delays.