Tan Son Nhat Airport is about a 15-minute drive to downtown (District 1). Metered taxis are plentiful at the airport. Figure on US$5 for the ride, which you can easily pay in dollars if you haven't changed money yet.
Taking a metered taxi from downtown to the airport costs around $5. (There is a legitimate 5,000 VND charge over the meter.) Vietnam Airlines will tell you to arrive at the airport two hours prior to departure for domestic flights. We usually leave about one hour before departure and have more than adequate time. For international departures we usually allow two hours. (Then sit in the departure lounge for 90 minutes wondering why we left so early, but you never know!) Departure tax is US$14 for international flights payable in dong or US dollars.
Saigon Train Station is off Cach Mang Thang Tam in District 3, near the traffic circle at Dai Lo 3 Thang 2. From there it's a short taxi or cyclo ride to District 1. Beware! Train travel in Vietnam will try the patience of a saint; it won't save you much money either. The two-hour flight to Hanoi costs $158; the 34-hour train costs $144.50 for an air-conditioned sleeper. Unless you're a train buff with plenty of time to kill, skip the rails. (Train buffs without time to kill may want to try the Danang-Hue Express.)
If you're traveling by bus, you'll be lucky to arrive at all! Even hard-core travelers who have traveled by bus in other developing countries should think twice before getting on a bus here. Enough said. Saigon has two domestic bus stations. The Mien Tay terminal, serving the Mekong Delta and points south, lies about 10km southwest of downtown. The Mien Dong terminal, serving points north, is to the north of the city on Xo Viet Nghe.