Before approaching a bulbdog, Make sure you have aquatic mushrooms. The amount varies between levels. All you have to do it out the mushroom into your inventory slot and approach the animal. Press Y ( On Xbox) to feed it. There should be a bar that shows its taming level. On a lystro if you walk to far from it, The taming restarts ( At least it did for me ) I am unsure if it is the same for bulbdogs. But better safe then sorry . . !!
Breeding them is fairly easy. The gestation period is about 3 hours. Once they are born you have to force feed them non-poisenous mushrooms till they reach about 1.0%. Their baby cycle moves pretty quickly. About 2.5% per hour est.