There's no train connection with the airport.
Car and bus
Ho Chi Minh City Airport is situated in the city's western suburbs, at only 6 kilometers from the center. From the city center you can reach the airport via the Nam Ky Khoi Nghia Road which turns into Nguyen Van Troi Road which leads directly to the airport. The way to the airport is clearly signposted. Keep in mind though that traffic in Ho Chi Minh City can be very busy. At quiet times it takes you 20 minutes from the center to the airport; at rush hour travel time can be more than one hour.
Two buses run to Ho Chi Minh City; a ride to the center costs 3000 dong. When traveling in a group, a minivan can be cheaper; you do have to negotiate about the rate though.
Car rental Ho Chi Minh Airport
Officially there are no car rental companies present at the airport. There are rental companies in the city. In and around the terminal you'll be approached by people renting chauffeur-driven cars. The reliability of these people is often debatable. If you accept, you'll have to seriously bargain about the price.
There are a vast amount of official and unofficial taxis outside the terminal. Two reliable companies are Vinasun Taxi (Tel. +84(0)8 272 727) and Mai Linh Taxi (Tel. +84(0)8 38 262 626).
There's a parking lot opposite the terminal entrance. You can safely park your car here for a short period, but don't leave it here at night. The airport frankly states that it takes no responsibility for cars parked here.