"ехать" means to be in progress of going somewhere (by car, bus, train etc).
"поехать" means to start going somewhere. (Maybe you remember the famous Gagarin's phrase "Поехали!" ("Let's go!") when his spaceship was about to take off).
"доехать" means to successfully finish going somewhere (to arrive somewhere as a result of going somewhere).
When you say "I need to go to airport" to taxi driver, you mean that you need to actually arrive there, so you should use "доехать". "Мне нужно поехать в аэропорт" sounds a bit weird in this context.
Although in other contexts, not related to the process of going somewhere, when departure and arrival are not considered separate actions, "поехать" can be used.
"Я должен поехать туда во что бы то ни стало." - I must go there by all means.