Goku learned instant transmission on the remote planet Yardrat. Many others were capable of performing this technique, none of which were relevant to the show and thus not shown.
The only one, other than Goku, that was shown using the technique was Cell. Since Cell was partly made out of Goku's cells, he could sense how it needed to be performed by seeing Goku do so.
For the others, nobody just bothered to go to all the way to Yardrat to learn the technique. Why they didn't ask Goku is beyond me though. I guess it wasn't needed for the story and Akira didn't want anyone else to look as bad-ass as Goku.
Other forms of instant transmission were seen though.
Fortuneteller Baba and Kami only use this technique to transport people from the Other World to the living world. Mr. Popo, assistant to the Guardian of Earth, can use a version of this technique while sitting on his Magic Carpet.
Kibito uses this technique after being resurrected to find Supreme Kai and Gohan, and take them to the Sacred World of the Kais. He uses it again to take Gohan back to Earth after the latter had his full potential unlocked by Old Kai.
Kibito Kai, who gathered this technique from Kibito when Kibito and Supreme Kai fused, uses this technique on many occasions: he used it to save Goku, Vegeta, Dende, Mr. Satan and Bee from being killed when Kid Buu blew up Earth, and later, in Dragon Ball GT, to save the whole population of Earth from the planet's explosion.
Kid Buu using Instantaneous Movement
Kid Buu learns this technique by watching Kibito Kai perform it. In the anime, his technique is referred to using the generic name Instant Transmission.
(Non-)Canon user list
For a full (non-)canon list, see the full list here