Motorbike

Crossing the road in Saigon can be a nightmare. The trick is to disconnect the part of your brain that processes fear. Be Moses. Walk slowly and confidently - the sea of motor scooters will part every time. After this, you’re ready to for some serious fun - a trip on a motorcycle taxi (xe-om or moto for short). Whilst a few vice peddlers at the bottom of Dong Khoi give these chaps a bad reputation, and few speak much English, it’s the only way to see this town. Note that your moto driver will be getting a kickback from some places he’s recommending. He needs it to live, so if you get a good moto, tip well. If not, stop at the nearest corner and duck into a shop until he’s gone. 20,000 for short trips, 150,000 plus tip for a half-day. DO NOT USE A CYCLO unless the hotel is paying, they will cheat or rob you. 

If you decide that you want to experience riding around the city yourself, you can hire a motorbike  for about 10$/day. Go to Pham Ngu Lao - De Tham area, there are plenty of shops there. Riding a motorbike requires a Vietnamese driving licence, if you get a visa longer than 3 months, you are eligible for one. However, in monsoon season, parts of the city like District 6, District 7, Binh Thanh will be flooded after a heavy rain.

Taxi

Taxis are also plentiful, cheap and a much safer bet late night. Read here or here for airport trips.

Maps

Most hotels will provide a free tourist map of district 1 although these vary in quality and tend to be advertising based. The Sheraton have one of the best of these and will provide one if you ask at reception. Within District 1, 'Bookazine' at #28 Dong Khoi (between Ngo Duc Ke and Ho Huan Nghiep) have larger city maps if you plan to venture beyond District 1. The one published by Du Lich & Giao Thong has a street index on the back. Fahasa Books also carry a full range of available maps. They have two large stores in District 1 - 185 Dong Khoi, just down from Le Thanh Ton, and 40 Nguyen Hue, just down from Mac Thi Buoi. MySherpa Travel has also published tourist maps of central District 1 with all shops and points of interest marked. Outlets in Saigon include Gaya, Dolce Casa, Annam Fine Foods, T&V Tailor, Galley Deli, and a number of two star hotels. Also see hochiminhcitymap.info.

Maps for iPhone

If you have an iPhone "offmaps 2" is a great app to have by your side. For around $7 you can download unlimited maps for many different cities and towns that operate offline (without data usage, so you could use your IPhone without a sim as long as it has GPS enabled).
So you can set a point where your hotel is and, as long as you have battery on your iPhone, you can find your way back after wandering about all day or use it to find your way easily to places you want to visit. You can also download offline city maps with the TripAdvisor app.


Walking tour
(see details for each attraction):

From Saigon River, walk to Nguyen Hue Street in District 1, at the end of the street is the building of the People's Committee of Ho Chi Minh City. Entry is not allowed however you can take photos in the small park in front of this building. It appears that this area is one of the most well known spots of the city with buildings, hotels, shopping malls, trees, flowers, people and vehicles, etc. During the biggest festival in Vietnam (Tet - Lunar New Year), Nguyen Hue is the amazing flower walking street.

Walk back a little from Nguyen Hue Street, turn left to Le Loi Street, you will see the City Theater (City Opera House), intersection of Le Loi and Dong Khoi Street. On big festivals such as Tet, Victory Day 30 April, National Day 2 September, live bands perform at the open area in front of the Theater and people gather to watch.

Walk on Dong Khoi Street against the one-way direction and  you will eventually reach the Notre Dame Cathedral. It is very crowded inside and outside during Christmas night time (December 24 and 25).

On your right hand side, there is the Central Post Office.

Continue with the Dong Khoi Street, after passing the Notre Dame Cathedral, the same street is named Pham Ngoc Thach. Walk straight, you will see Le Duan Street. Turn right and walk to the end, intersection with Nguyen Binh Khiem Street, the Zoo and Botanical Garden (Thao Cam Vien) is there. There is Vietnam History Museum inside.

Go straight from the zoo entrance to the other end of Le Duan Street, intersection with Nam Ky Khoi Nghia Street, you will reach the Reunification Palace (Thong Nhat Conference Hall, Dinh Thong Nhat, Dinh Doc Lap). There is a park in front of this Palace.

Going out from the Reunification Palace, turn right and walk on Nam Ky Khoi Nghia Street towards Saigon River direction, on your left you will see the City Museum near Ly Tu Trong intersection with Nam Ky Khoi Nghia Street. Walk straight and turn right to Le Thanh Ton Street, walk for a few blocks, you will see Ben Thanh Market.

Or going out from the Reunification Palace, walk on Nam Ky Khoi Nghia Street, away from Saigon River direction until you reach Vo Van Tan Street, turn left again and walk straight to the War Remnants Museum (Nha Trung Bay Toi Ac Chien Tranh) near Le Quy Don Street.