Taran makes an ass of himself #1: Upon meeting Prince Gwydion for the first time, Taran manages to spectacularly put the wrong foot out in every instance. First he exclaims that this man before him can't be Gwydion since he's not dressed all prince-like. When he realizes it actually is the Prince, he tries to impress him, but only manages to nearly drown, dive into a thornbush and make a multitude of other ridiculous mistakes. The entire time Gwydion gently chides him and teaches him valuable lessons that stick with him for the rest of the book.
Taran makes an ass of himself #2: Gwydion and him get thrown into the Spiral Castle where he soon meets Princess Eilonwy, a ragged girl who lives in the castle. From the very start, Taran's penchant for dramatics clashes with Eilonwy's practical nature which results in some very hilarious exchanges. And a lot of comments questioning the intelligence of certain Assistant Pig-Keepers. Eilonwy helps him escape and rescues the other man in the cells which Taran assumes is Gwydion. It's not. It's Fflewddur Flam, wandering bard/king. Taran flips at Eilonwy who points out that his need for secrecy caused the whole misunderstanding in the first place.
Taran makes an ass of himself #3: So, they depart to Caer Dathyl and finally get Hen Wen back along the way. Their band, now consisting of Fflewddur, Eilonwy, Gurgi, Doli and Taran as the leader tries to head off the Horned King's men and warn the Sons of Don before it's too late. Unfortunately, they did get there too late and bands of warriors begin overtaking them for the fight. The Horned King himself chases after them and Taran draws the sword Eilonwy protected, the one he had no business pulling, and gets himself burned and knocked out for the rest of the fight. Thankfully, Gwydion came to the rescue and saved them all with the sword, defeating the Horned King.
Taran's many foibles turn what could be a very prideful egotistical character and humble him greatly. By the end of the story he's turned into a respectable guy, who realizes all the trumped up dreams and fantasies he had in his head aren't quite as important in a real quest. What makes him truly a great character is that not only does he screw up a lot of times, but when he screws up he learns. While he's always stubborn, he does take advice from Gwydion to heart, remembering it at different times throughout the book and acting on it. And after burning himself on the Book of Three and the sword Dyrnwyn, he's stopped touching the untouchables too.