Not far, although BOG is a big city, so it would depend on what part of BOG you wanted to visit. Some of the most interesting parts of BOG are less than half an hour from the airport by cab.

The centre of the city's old part is Plaza Bolivar. It's worth seeing. From there you could stroll around the Candelaria which is really pleasant. There are a number of great museums in the area and nearby (the Gold Museum, the Botero, la Casa Moneda), though if you're between long periods of being in a metal tube, you might prefer to stay outdoors.

The more modern parts of Bogota are the Zona Rosa and the Parque de la 93. The Zona Rosa is an area of shops and restaurants. Parque de la 93 is a small park with a bunch of restaurants built around it. It's quite attractive.

The city's tourist authority has a booth at the airport. You'll pass it as you're about to leave the customs hall. If you decide to pick up a travel guide for Colombia, I'd definitely go with Lonely Planet as it's very up-to-date. The Frommers guide must have been written during the height of the city's security problems. I found it next to useless.