This page explains how to start and complete the The Weapon Connoisseur side quest.
How to Start the Quest
Talk to a boy named Nebb in Hateno Village, which is east of Hateno Tower, to start the quest.
How to Complete the Quest
Nebb's first request is to see a Traveler's Sword. The Bokoblin camp between Great Plateau Tower and Oman Au Shrine has a locked treasure chest that contains a Traveler's Sword. Defeat all the Bokoblins in that camp to unlock the chest and get the sword. Then go back to Hateno Village, equip the sword, and talk to Nebb.
Next, he asks to see a Fire Rod. These are wielded by Fire Wizzrobes. One place to find a Fire Wizzrobe is between Crenel Hills and Crenel Peak, which are north of Dueling Peaks Tower. Once you have a Fire Rod, go back to Hateno Village, equip it, and talk to Nebb.
Afterward, he asks to see a Moblin Club. The Moblins north of the Great Plateau wield them. Once you have a Moblin Club, go back to Hateno Village, equip it, and talk to Nebb.
Then he asks to see a Duplex Bow.