In The Taming of the Shrew, Tranio is the servant who spends much of the play pretending to be his master, Lucentio. Grumio is Petruchio's servant, who serves his master faithfully, though ineptly. Which of these is the better servant depends on your definition of "better." Tranio is a character derived from Greek and Roman comedy, Plautus in particular, where one of the stock characters was that of the slave who was cleverer than his master. It is Tranio who shows Lucentio how to get close to Bianca. In doing so he serves his master well, though Bianca's headstrong nature might mean that the marriage does not turn out as either the master or the servant planned.
Grumio is a comic character who is forever misunderstanding Petruchio's commands, beginning with a long exchange in which his master tells him to knock at a gate, which Grumio interprets as an order to hit Petruchio. Grumio would have been played by the clown of the company, with a great deal of comic exaggeration. There is no danger of his exploiting Petruchio, but he does infuriate his master with his constant antics and gross incompetence. Which is the better servant, then, depends on whether you would rather have an intelligent and resourceful servant who may pursue his own agenda or a faithful but foolish one who has not the wit to do anything but whatever you tell him to do, in the most literal way possible.