Cologne is well serviced by an extensive tram system. Tourists who are only "stopping through" would do well to purchase the 24-hour Welcome Card, which provides a day of limitless travel and discounts for various attractions in the city. Group day-tickets are also available. But, as always, the cheapest option is traveling by bike, and Cologne's size is manageable enough on two wheels.
There is no cheaper or more ideal way to see the city than by bike. Germans are always very good about marking bike lanes and often having them on the sidewalk, out of the automobile traffic. There are countless rental options all around town (such as Colonia Aktiv), and hotels and hostels often have bike-rental services though often pricey. There is also the old Deutsche Bahn Call-A-Bike standby, which requires online registration for free access, though with time-limits, to bikes stationed around the city.
Though it's technically not a metro, Cologne's tram system nonetheless goes underground and is just as extensive a subway system. East and West of the Rhine is solidly covered, and the ticket prices are not too steep. For tourists, lines 3 and 4 (North-South), 15 and 16 (West-East) will come in handy for trips into the Altstadt and Ehrenfeld. Protect yourself from high fines by not attempting to ride illegally ("Schwarzfahren"), and make sure to validate the ticket in the "punchers" either in the station or on the tram.
Cologne has a decent bus system that reaches some areas that the tram system doesn't, but for travelers who are only in the city for a short period of time, it might be best to ignore this option and focus on learning the none-too-intuitive tram system. The special Welcome Cards and day-cards also can be used on bus transport.
Taxis and Uber rides are always on hand, with drivers who almost always speak very good English, for those who want to expend less energy on navigating the city on their own.