Check out "From Pusan to Panmunjeom" by Paik Sun-yup. Paik was a colonel in charge of the ROK First Division, responsible for the defense of South Korea from the Yellow Sea (above the Ongjin Peninsula, quite a bit further west than the current border) to the edge of Paju/Dongducheon. By the end of the war, he had been promoted to full General (4 stars), the first four star general in the Korean military. There are two things to note about this book. First, it's written by Paik himself. He was in charge of a division at the outbreak of the war, he became the commander of the ROK I Corps, he commanded a special operation, Operation Rat Killer, to root out communist guerrillas in Jirisan in the end of 1951, and he became the Army Chief of Staff in 1952. You have to read everything he writes with that in mind. He glosses over massacres committed by the ROK and US forces with only a few sentences. To be fair, the book isn't about the massacres, it's about his role in the war, but he was a government man. He worked for the Park Chung-hee administration as an ambassador and is a military man. The second thing is that the book has a glowing forward from Generals Ridgway and Van Fleet who both speak quite highly of Paik, who in turn speaks highly of them. I'd say that does give it some authority that it's accurate, but still, read it with a grain of salt.

As for the comment, yes, in the spring of 1951, it would be fair to say that the ROK army ran when the Chinese attacked.

Against the North Koreans, the ROK forces held their ground as best they could. The North Koreans were better trained and better equipped, but Paik and his First Division held the North Koreans from Seoul for three days. To give you an idea of how much we're talking about here, the ROK First Division had just under 10,000 men in it, but at the time the war broke out, they had about 5,000 on the line. The North Koreans attacked with the NKPA First and Sixth Division and the 203rd Armored Division, totaling 21,000 men. With a 4:1 disadvantage, the First was able to hold of the North Koreans from reaching Seoul for 3 days, a distance of about 25 kilometers. On June 29th, the First Division crossed the Han River and regrouped at Siheung. MacArthur flew in and visited with the Corps commander at the time, General Kim, and told him to hold the line for three days to give US forces a chance to arrive from Japan. They ROKs held the line at the river for six days.

Against the Chinese it was a different story, however. Paik writes in his book about a paper, "Military Lessons Bulletin" from November 20, 1950. He got it from a Chinese POW and it was written by Teng Hua, Deputy Commander for the PLA. On the US Army, he wrote:

the US Army relies for its main power in combat on the shock effect of coordinated armor and artillery... and their air-to-ground attack capability is exceptional. But their infantry is weak. Their men are afraid to die, and will neither press home a bold attack nor defend to the death... Their habit is to be active only during the daylight hours. They are very weak at attacking or approaching the enemy at night... If their source of supply is cut, their fighting spirit suffers, and if you interdict their rear, they withdraw on their own.

On the Korean Army, Teng Hua said:

The Korean Army is deficient in all pertinent aspects. Their training is absolutely insufficient. Three Korean divisions do not equal one American division when it comes to firepower and overall combat strength. But the Koreans do have a measure of fighting spirit.

The Chinese used the knowledge they gained from their initial battles in October and November of 1950 to shape their strategy as the war went on. The main thing was to avoid fighting American forces and focus on the Korean forces. The Chinese would attack at night and they would play loud horns and beat drums and gongs and make a terrifying racket. When the attacks would commence, the Korean forces would give under pressure. The most notable example of this was during the Chinese Spring Offensive. The Chinese pushed hard in the eastern sector and during the attack, the ROK III Corps broke down and fled. Once the battle was over, General Van Fleet flew to the command post personally and disbanded the III Corps. The ROK Army Headquarters was no longer in charge of any combat operations, and the ROK I Corps was placed under the command of the Eighth Army. It was a major humiliation for the ROK Army. In traditional Asian fashion, many people resigned and were replaced, including the Army Chief of Staff, Chung Il Kwon.

It was around this time that training became a major issue. General Van Fleet established the Field Training Command in July 1951. 150 US officers and NCO's were brought in to train the ROK soldiers. By the end of 1952, all ten ROK divisions had gone through training. They found that units that underwent training suffered 50% fewer losses in men and equipment than untrained units. In December 1951, a Staff School was created in Daegu and in January 1952, a four year military academy opened in Jinhae. All the while, upper level Korean officers were flying to the US to attend classes at Ft. Benning, Ft. Sill, and the Command College at Ft. Leavenworth. Paik Yun-sup was in Leavenworth when he was recalled to Korea by President Rhee when he attempted to sabotage the armistice talks by releasing the POW's.

To conclude, in May of 1951, to say that the ROK forces would run from the Chinese is accurate. By the end of the war, the ROK had trained and been equipped quite well and controlled approximately 2/3rds of the front and had become a much more formidable force.

For sources, check out the book I mentioned, "From Pusan to Panmunjeom." For a Chinese take on it, if you can find it, "Memoirs of a Chinese Marshall" by Peng Dehuai has his views on the Korean War. But, good luck finding it - it's almost 50 years old. For the unit numbers and levels, I use "The Korean War: Vol 1" by the Korea Institute of Military History. Fantastic book, but it is very very detailed.