In videos of astronauts walking on the Moon, it seems very hard to move around with any efficiency, and it seems impossible to run.
Why is this? What could be done to mitigate these effects?
More than anything, the reason for this were cumbersome EVA (Extravehicular Activity) suits used with the six Apollo program missions that landed on the Moon. They were basically a one-piece suit, each made to fit (custom tailored) each of the astronauts.
There were two basic configurations of the space suits used to support Apollo Missions:
We will be more interested in the latter, the Apollo EV A7LB PGA EV suit to answer the question though, since it was the suit used for extravehicular activities on the Moon's surface by the 12 moonwalkers. Here is a photograph of how it looked like:
So we mentioned they were custom tailored suits, but they also look pretty cumbersome on their own, let alone with the heavy Primary Life Support System (PLSS) with a separate Oxygen Purge System (OPS) mounted on top of the PLSS, both strapped on the astronaut's back. While the gravity is of course a lot smaller on the Moon compared to sea level gravity on the Earth (roughly 1/6), PLSS still weighed 6.4 kg (14 lb) and the OPS 3.1 kg (6.8 lb) on the Moon. That's additional 9.5 kg (20.8 lb) of bulk that each moonwalker had to carry around.
But that's not the whole story. The EVA suit itself weighed 34.5 kg (76 lb) on the Earth, translating to roughly 5.7 kg (12.6 lb) on the surface of the Moon, and we still didn't add the helmet, and the boots, and any other equipment standard on Apollo lunar EVA to the bulk, and all of that amounted to a total of roughly an additional weight of 18-20 kg (40-44 lb) that needed to be carried around at all times during the lunar EVA.
These roughly 20 kg (44 lb) might not sound as much, but try working with this bulk on your back for several hours, and you'll soon appreciate the weight. Quite literally. And that doesn't even account for the weight of any samples astronauts took from the lunar surface and needed to carry around in sample bags before returning them to the Lunar Module during each day's EVA. So while Apollo moonwalkers were rather happy with how their space suits were, with Neil Armstrong being quoted saying his Apollo 11 A7L space suit to be "tough, reliable and almost cuddly", they were also cumbersome to work in for longer periods of time, and astronauts weren't able to keep up with their packed tight schedules.
Of course, as Loren Pechtel commented, it's not merely the weight of the EVA suits in Lunar gravity, but their mass too. Inertia and Newton's first law of motion will prevent the suit to change speed and direction without a sufficient force to move its mass where the moonwalker in it intended to move. So they might weigh less, but they still have (more or less, they did get rather dusty and PLSS/OPS were not closed-loop systems) same mass as they did on Earth.
NASA soon realized this and included the Lunar Roving Vehicle (LRV) in the last three missions of the Apollo program (Apollo 15-17) between the years 1971 to 1972 to help moonwalkers traverse distances needed to complete their scheduled duties:
LRVs were used for greater surface mobility during the Apollo J-class missions, Apollo 15, Apollo 16, and Apollo 17. The rover was first used on 31 July 1971, during the Apollo 15 mission. This greatly expanded the range of the lunar explorers. Previous teams of astronauts were restricted to short walking distances around the landing site due to the bulky space suit equipment required to sustain life in the lunar environment. The range, however, was operationally restricted to remain within walking distance of the lunar module, in case the rover broke down at any point. The rovers were designed with a top speed of about 8 mph (13 km/h), although Eugene Cernan recorded a maximum speed of 11.2 mph (18.0 km/h), giving him the (unofficial) lunar land-speed record.
The LRV was developed in only 17 months and performed all its functions on the Moon with no major anomalies. Scientist-astronaut Harrison Schmitt of Apollo 17 said, "The Lunar Rover proved to be the reliable, safe and flexible lunar exploration vehicle we expected it to be. Without it, the major scientific discoveries of Apollo 15, 16, and 17 would not have been possible; and our current understanding of lunar evolution would not have been possible."
So there we have it, hopefully answering all your questions, except maybe why none of the moonwalkers was seen running. Not an official explanation at any length, but I'd presume Apollo astronauts not to be of the easily scared variety, and not inclined to running for no good reason. There was a fair deal of jumping going on on the lunar surface, though:
On a more serious note though, the PLSS and OPS systems had performance parameters that meant they will only be capable of sufficiently supplying oxygen, cooling down and circulating the air inside the space suit at a certain rate, to prevent the astronaut from overheating, excessive sweating, and using up the oxygen supply too quickly.
Even if one of the astronauts tried running, they would most likely be advised against it by their Flight Director, a person they would most likely obey, if for nothing else then out of respect for Gene Krantz, one of the Apollo program Flight Directors that also directed team efforts to save the crew of Apollo 13 and was in that position also for the first landing on the Moon by Apollo 11 Lunar Module (LM) named "Eagle", on July 20, 1969.
How to fix this? Newer materials and tech could lighten a suit somewhat. There's also research into suits that don't use air-pressure, but instead strap-around the body like a compression-garment to counter the vacuum. (There's air for breathing of course)
Moving in low-gravity is something we don't have a practical fix for. Practice in actual low-grav is about the only way to do it, and it's hard to get to the moon to practice.
As far as the lunar surface goes, we're thinking of paving the place. (No really: http://arc.aiaa.org/doi/abs/10.2514/6.2005-2510 )