Dragonball Z Training
Vegeta's Training
Not a lot is known about Vegeta's training when he was younger, but from the Bardock Special it is evident that he went through extensive training against saibamen, able to defeat whole groups of them as a small child. He also conquered a lot of planets for Frieza, and probably got a lot of strength power-ups from taking serious damage. I assume that he must have had the same basic training as most of Frieza's troops, which is most likely to be his foundation.
King Kai's Training
This is quite possibly the most influential training of anyone's life. To reach King Kai, you must run Snake Way, and that is no easy feat. It took Goku about 3/4 of a year to do it, and he was comparatively strong at that time! The thing that makes King Kai's training so good is not only the obscure methods he uses, but the increased gravity on the planet means that the persons strength and speed will be increased. At first King Kai taught Goku strength and speed by getting him to chase Bubbles, and hit Gregory. Once this was accomplished, Goku went on to learn the two most useful attacks in his life:- The Kai-o-ken, and the spirit bomb. These were useful allies, and quite possibly the best attacks in the whole series. When leaving King Kai's, Gregory states that he's even stronger than the master himself, which is quite an achievement.
King Kai has also trained Yamcha, Tien, and Chao-zu, but he only taught them the basics of strength and speed, as none of them were capable of learning the attacks that Goku did. He attempted to train Piccolo, but the Namek decided to train by himself, but was a bit disappointed when he saw how fast and strong everyone else had become from King Kai.
Piccolo's Wilderness Training
After Goku's death, Piccolo took Gohan, and decided to utilize his almost unlimited potential. At first Piccolo just left him in the wilderness to survive on his own, teaching him to show courage, and making him a lot faster and stronger. Piccolo slowly formed a bond with the kid, feeding him, and watching over him, becoming one of the good-guys. He later returned, and personally trained Gohan, giving him a good foundation in martial arts. Piccolo's training was tough, harsh, and somewhat painful, but Gohan pulled through, and became an excellent fighter, only really able to show it during the end of the Saiyan saga. It didn't take Gohan long to become stronger than Piccolo, achieving it after his spirit and time training with his dad.
Piccolo's Self-Training
This time around, Piccolo was training himself in order to be strong enough to combat the Saiyans, who's arrival was imminent. He meditated, and increased his power greatly, especially with the use of his mind, able to lift entire pyramids without touching them. Gohan was amazed that Piccolo was able to achieve such seemingly impossible feats.
Kami's Training
This differed greatly to the training he gave in Dragonball, with him having a group of the most powerful warriors in the world on his lookout this time around, and teaching them very little. He helped them increase their power slightly, but compared to Goku, their advances were insignificant. The one crucial facto of this training was the room of spirit and time, where everyone found out how impossibly strong Saiyan are, and that it'd be an extremely tough fight to defeat them.
Increased Gravity Training
Goku was the first person to use this, training intensively hard on his way to Namek in order to have the power to save his friends. He accidentally got a lot powerful by hurting himself, but didn't abuse it due to limited sensu beans. During this time he pushed himself way past the limit, and discovered how powerful he could become. He got the idea of gravity training after he'd been to King Kai, and had such an increase in power there. The main thing that improve was his speed. He managed to get the dial all the way to 100 times the Earths gravity by the end of his trip.
Vegeta also used a capsule pod for his gravity training, trying to become strong enough to defeat both Goku and the androids. He got powerful enough to reach the entry point to become a Super Saiyan. He managed to go all the way to 300 times the Earths normal gravity. Chibi Trunks also trained in here, with Vegeta teaching him, and helping him become a whole lot more powerful in order to do well in the martial arts tournament.
Saiyan Ability To Increase Power When Close To Death
Whilst not essentially a form of training, this was used to make the Saiyans a lot stronger. When healed after almost dying, a Saiyan can almost double their power. Goku accidentally used this, but Vegeta discovered this during the battle against Frieza. He abused this slightly, purposely getting hurt in order to become stronger, but it wasn't quite enough to defeat Frieza, although it gave him a crucial power boost.
Yardrat Training
When Namek blew up, Goku managed to escape in a pod, and landed on a planet called Yardrat. Here he met some very friendly people, and whilst they weren't very strong, they knew lots of useful techniques. It was here that Goku learn instantaneous movement, a skill that later proved vital in getting him out of some tight situations, and saving the Earth too.
Spirit And Time Training
This was critical in assisting the warrior in terms of power so they could defeat Cell. Two people went in at a time - First it was Trunks and Vegeta, then it was Gohan and Goku. Whilst the former people went all out, and managed to surpass the level of Super Saiyan, Goku and Gohan decided to increase their power as Super Saiyan, as ascending makes you a lot slower. Everyone got infinitely more powerful, but it was Gohan who benefited from it most, with Goku forfeiting a lot of the training he could do in order to help his son get stronger.
Future Gohan's Training
This was the base style given to Future Trunks, as Gohan taught him to fight in order to beat the androids. This happened in an alternate timeline to the one we are familiar with, and it increased Trunks' strength massively. When Gohan died, Trunks went past the limit of anger, and became Super Saiyan, a feat which would have never been accomplished without Gohan's help. Admittedly, Gohan's training was completely original, he preferred to train Trunks in a similar way as Piccolo trained him.
Afterlife Training
After sacrificing his life to stop Cell from blowing up the Earth, Goku was stuck in the afterlife, but that didn't stop him from training and becoming stronger. He fought against many warriors from the past, and was trained some more by King Kai. He learnt the skill of fusion, and how to become SSJ 3 whilst dead. This contributed greatly to his knowledge of fighting, and his technique, as well as hugely increasing his strength. His improvements were put to the test when he was wished back to fight in the martial arts tournament, with it being interrupted, and him going onto fight Majin Vegeta and Buu, his strength proving vital to his overall victory.
***More coming as soon as I have seen the rest of the episodes, or read the rest of the manga.