An entire book could be written on the training of sumo wrestlers (infact they have). Here, I will give the brief, condensed version. Training takes place at what are called stables. The stable is where a sumo wrestler will live until they reach a certain level when they are then allowed to travel out or live away from the stable. This is usually after several years, I believe something like 5 or 7 years if I recall correctly.
Anyways, the junior guys start at the bottom. They are responsible for cooking, cleaning, massaging and training with the more senior guys. The cooking part is very important because not only do the sumo wrestlers in stables eat a lot, but many of them are also excellent Japanese chefs after their years of cooking in the stables. Many sumo champions go on to open their own restaurants after they retire from wrestling.
As for the training it consists of wrestling drills, technique and repeated practice. There is also a high emphasis placed on flexibility. These guys might be huge but they can all do the splits. They can all sit on the tatami with their legs open and touch their chest, forehead, nose, etcetera to the tatami. There are also different weight classes for sumo wrestlers, so although they are all trying to get big, not all are as huge as the famous ones we have all seen.
This training last a few hours in the morning and a few hours after a long lunch and nap. Yes a nap. It is one of the techniques used to get these guys big. A huge lunch of usually chanko nabe in the order of 9,000 to 10,000 calories is standard. Then comes the beer. They get to drink a lot of beer right before their nap. The nap is about 2 or so hours. Eating and sleeping after a huge meal is a great way to put on weight, and the beer helps. After the nap comes another round of training, stretching, and massage.
The cool thing is, once a stable decides to take you on you start to earn money, through a small stipend (again if I recall correctly) and more money through pre arranged matches to get you ranked and move up the ladder to progressively more bouts and higher pay. Since for the majority of your career in sumo you live at the stable and cannot leave, not even to go shopping or visit the sights of Japan, sumo wrestlers have a pretty decent bank account when at higher levels they do get permission to depart the stable or get their own apartment. Then of course winning champion matches earns them even more money.
Just as a side note, there have been Brittish and American wrestlers accepted into stables for training, so it is not strictly limited to Japanese. This is Akebono Tarō (Chadwick Haheo Rowan) a sumo champion from Hawaii.