They may have beaten the Taira, but the Minamoto soon fell victim to the same ploy they were using on the emperor. After Yoritomo died in 1199, his widow's family (the Hojo clan) exercised power by always providing regents for the shoguns. Thus the head of the Hojo clan would tell the shogun what to do (in very polite language), the shogun would then tell the emperor what to do or sign or say (again, in very polite language). This setup lasted until 1333.
During the roughly 150 years that the shoguns lived in Kamakura, several important events took place. The most interesting were the Mongol's two attempts to invade Japan. The Mongols started their conquests in 1211 against neighboring horse peoples before joining the big league by taking on and then taking China. By 1258 they had taken Baghdad and were beginning to worry the Christian crusaders and the Middle Eastern Muslims. In September of 1260, the Mongols clashed with the Mameluke army of Egypt just outside Jerusalem and for the first time, the Mongols lost. After that battle, Mongol expansion into Europe and the Middle East was over, but they continued to go south and east. In 1274, for the first time in history, Japan was invaded, and for the second time in history, the Mongols lost. They lost again in 1282 when a typhoon destroyed their invasion fleet. The weather played a role in the repulse of the Mongols in 1274 as well, but both times the samurai had to fight and they held their own both times until the typhoons came and swept the invaders away. Relieved Japanese believed that the gods had sent the typhoons to protect their islands and this is the origin of the ``kamikaze'' myth which played a small but painful role in the Second World War.
Two other important events during the Kamakura period involved the imperial court. The Hojo were able to consolidate their power after an emperor attempted (with a spectacular lack of success) to regain control of his government in 1221. Then, between 1333 and 1336, the Emperor Godaigo actually managed to rule more than whatever room he was in. He did this with a lot of help from his friends--friends who didn't like the Minamoto / Hojo government. When one of these `friends' turned against him in 1336, Godaigo fled south to the Kii mountains and set up a court there. The disloyal friend, Ashikaga Takauji, merely put another member of the imperial family on the throne and went ahead with the business of ruling in the name of the ``emperor.'' Eventually, the southern court was lured back to Kyoto by promises that the two branches would alternate on the throne. That was in 1392 and the southern branch is still waiting for their turn. (Actually, I think they gave up on ever getting their turn long ago.)