Kaio-Ken style:

Kaio-sama's (King Ki) training style is initially similar to Muten Roshi's training, where it increases Goku's
strength and speed. But once that is done, Kaio teaches his martial arts style to Goku. Although little is seen of this in the series, there is no doubt that Kaio can fight. Once Goku finished the training he takes with him two attacks: the Kaio-Ken and Genki-dama. Goku's body was only able to sustain the Kaio-ken at level X2 at this time. Kaio has also trained Yamucha, Tein-shinhan, and Chaosu, but it is believed that he only taught them to increase their strength, as such these warriors possibly were incapable of learning the true Kaio-Ken style.

Piccolo trains Gohan:

Piccolo inherited many of his techniques from his father (Elder Piccolo Daimao) and trained Goku's son, Gohan, under that style. Gohan learned to fight much like Piccolo and inherited some of his moves and techniques. This is the base martial arts for Gohan, but he later alters it a bit after training with his father.

Vegita's extensive training:

Although not much is know about the "style" and foundation of Vegita's martial arts, he underwent many extensive training sessions, presumably from the Saiya-jin's and Freeza's technology. He also traveled and battled from planet to planet increasing his strength and prowess.

Training in the 100X-300X gravity capsule:

Goku trained on his own while heading to Nameck in a spaceship that can increase the gravity level 100 times inside the ship. This enabled him to increase his strength, allowed him to expand the use of the Kaio-ken style and increase its level to over 20 times. Vegita also trained in this way and used it as an entry point to get to the Super Saiya-jin level. Saiya-jins' ability to double their power

level when recovering from near death:

During the battle with Freeza, Vegita made a startling discovery: every time a Saiya-jin is near death and recovers, their power level nearly doubles. An interesting "technique" used to increase power is to bring oneself to near death. Although Goku never used this purposely, Vegita used it many times and I guess you can say abused it.

Goku trains with the Yardarrat-jin:

Goku learns one strange, but useful technique from the people of Planet Yardarrat: Instantaneous movement (instant transmission).

Future Trunks trains with Gohan in the future:

The Trunks from the future learned many of his techniques on his own, but was briefly trained by the Son Gohan of the future. This is future Trunks foundation.

Saiya-jin training in the room of time (hyperbolic chamber):

The Saiya-jin's of earth (Goku, Gohan, Trunks and Vegita) trained in the room of time in order to increase and surpass the level of the Super Saiya-jin. Trunks and Vegita took one route, increasing their strength and power respectively and going to the next level of the Super Saiya-jin. Goku and Gohan however, stayed at the normal level of Super Saiya-jin and increased their powers at this level. Here Goku trained Gohan.

Goku trains in the afterlife with Kaio-sama:

Supposedly Goku was to train with the master of the Kaios, Dai Kaio-sama. However, it is rumored he feared that Goku would surpass him. So Goku never got to train under him. But that did not stop Goku from continuing his training. He continued to train with the warriors in Hades and under Kaio-sama. Here learned Fusion and how to transform to Super Saiya-jin 3 .

Vegita trains (young) chibi Trunks....

Gohan trains Goten, Videl and (young) chibi Trunks

Nothing much to note, Gohan took the time to train Goten and Trunks martial arts, while he taught Videl how to fly. Trunks has a foundation of both Vegita's and Gohan's training, while Goten only has Gohan as his foundation.
Goku teaches (young) chibi Trunks and Goten Fusion

Goku trains Trunks and Goten on how to do Fusion. He also showed them Super Saiya-jin level 3. And although he never
taught it to them, the two kids learn to mimic it during their training in the room of time.

Dai Kaio-shin trains Gohan:

Dai Kaio-shin used a special technique to make a Gohan a more powerful warrior. The result is a tremendous increase in power, speed and battle prowess (similar to the power up of a Super Saiya-jin, but without the anger and emotions). They train on planet Kaio-shin, far above the heavens.