What is the cheapest type of printer to buy and run?

It depends solely on how much you will use it. For example, laser printers are more suitable for offices, schools, and environments, where the printer is used almost constantly. If it's used a lot, you're probably looking at a laser printer, since the cartridges have a slightly higher price, but a much higher yield. If you can't afford a laser printer, and you're only going to use it at home, you're probably looking at an inkjet. If you need to print off something for school, ask your teacher, or someone that works at your school. Some secondary/high schools might allow you to use their printers at a lower price