There are several ways to get from A to B as a foreign student in Colombia.
I can really recommend buying a cheap bike. The campus is quite big and it took me about 15-20 minutes to walk to class every day. It also came in handy for buying groceries or visiting friends that lived nearby. There is a road somewhere in Bogota with about 20 bike shops lined up side by side. Unfortunately I do not remember exactly where it was (just ask friends in the university. Someone will know). I invested about 70 Euros for a bike and it definately payed off. But you should be cautious when you buy such a cheap bike. I would recommend to make sure that all screws on it are tightened and to try it out for a few minutes before buying it. In that price range you should not expect too much.
Taxis in Bogota are very cheap compared to taxis in Germany but compared to buses in Bogota they are quite expensive. You should be aware that it is not considered absolutely safe to just stop a cab on the street. In the past sometimes people were not delivered to the agreed destination. Instead the driver would then stop in a different place and let two accomplices into the cab. They would then drive to a cash machine and force the passenger to withdraw money and give it to them. However, the frequency of these incidents seem to have decreased greatly over the past years. The safe option is to call a taxi company (numbers are on the back of the cab. Something like 111 111, etc.) and to arrange a safe ride. The taxi company will then tell you the license plate of the taxi and a code word you have to tell the cab driver when he arrives. This ensures that he is trustworthy. But if you know what you are doing or are with people who know what they're doing you can also stop taxis from the street.
Transmilenios are buses that have their own lanes on the streets in Bogota. In principle they are similar to our S- and U-Bahn system. There are fixed transmilenio lines, stations and timetables and usually much faster than cars and private busses because they do not have to deal with the traffic jams. On the other hand they are often very crowded which makes them a perfect environment for stealing. There are actually police comercials advising the passengers to not wear there backpacks on their back but on their belly. To go from any Transmileniostation to another you pay about 50 cents. As long as you stay inside the bus stations you can change busses as often as you want. Only the entry into the bus station costs 50 cents.
There is a very confusing system of private buses operating on fixed routes in Bogota. In the front window of every private bus is a sign with some abbreviations from which one can deduce on which route the bus is operating. To get on the bus you must simply raise your hand and look at the bus driver. The bus will then stop and let you hop on. A ride also costs about 50 cents. During rush hour you will be stuck in traffic quite often. Again, one should be wary of pickpockets.