A Bicycle is a Key Item. The Bicycle allows for faster traveling than walking. It also allows the player to pass the Cycling Roads on Route 16 to 18 in Kanto.
In Kanto you need a bicycle voucher from Vermillion which you can exchange for a bike in the bike store in Cerulean (Guide).
In Johto you can get the bike easily without any requirements. Just go to the bike store in Goldenrod and you will get one for free. Also for this we have a small picture guide, so you can find it easier (Guide).
A large and rare mushroom. It is very sought after by collectors. Is needed at Egg-Tutors to learn new moves. Can be sold for 2000₽ to the Gourmet Maniac. Paras and Parasect in Kanto have a 5% chance of dropping it after a fight.
A blue shard of an ancient tool made long ago. The Blue Shard is needed as payment to learn moves from the Master Move Tutor in Lentimas Town in Unova. Can also be traded for berries at Juggler Domino in Violet City or sold to Ore Maniac in Unova for 200₽.