Durham is the type of place that you can't really navigate by foot, nor would you want to.  A rental car (or taxi) is really the best option for getting around town.  In general, the city is fairly well spread out, without one central "draw" where everything happens.

If you're coming in from the airport (~12 miles from Durham), you'll likely stay in one area of the city, visit Duke near the center of the city, perhaps go shopping down south at the new Streets at Southpointe Mall, grab a bite to eat near Brightleaf Square, then head over to 9th street for some drinks after dinner. The point above is that to really appreciate Durham, you'll need a car (or taxi) to get around to the different locations.   

Until you become familiar with the streets and street names, it's also very easy to be confused by the duplication of street names.  Sometimes it seems as if there are 32 different roads called "chapel hill road","erwin", or "cornwallis".

Durham is connected to the outside world by a few main highways:

Hwy 40 (that starts in Wilmington NC and ends in Barstow Calif) cuts through the southern part of Durham and heads out to the airport, then out to Raleigh.

Hwy 15-501 is the main road connecting Durham and Chapel hill. 

Hwy 85 heads North out of Durham up towards Richmond, then west towards Greensboro.

Hwy 147 - or the Durham Freeway - Connects I-85 through the middle of Durham to 40 - out near the airport.