Part 1

Requirements

Talk to Dr. Betty. She is on the western side of Ellinia.
Around the northern fields and forests of Ellinia there are some plants that can be destroyed by swinging your weapon. Killing these plants will occasionally give you Plant Samples. Collect 20 of them.
Return to Dr. Betty.

The plants don't always drop plant samples, and will usually give Red Potions and Tree Branches. Each plant requires 4 hits to be destroyed.
It is recommended that you go around the field and killing the plants. After some time, change channel, and repeat.
A good place is at Field Up North of Ellinia (from the town of Ellinia, take the north portal and you will be there), go to the far left of the map and attack the plant near the tree. Then change channels to another, and you will end in the same spot, hopefully near the plant. This method is useful if you do not wish to constantly be interrupted by monsters (although on some occasions, slimes will be lurking nearby). It also saves time so you don't have to wander around looking for a plant.
You can only successfully hit the plant when it is not vibrating. So every time you hit a plant, wait until the plant stops vibrating, then strike again. If you hold the attack button, the weapon may hit the plant while it's vibrating and thus you will waste more time killing the plant, so equip a weapon with a slow attack speed, such as a staff or wand. It is now possible to hold the attack button and deal constant hits to the plant.
Other good places are The Tree That Grew I, II and III. Just go up and down the tree searching for plants.

The Stump Teardrop can be obtained from Dark Axe Stumps
Weighted Earrings can be obtained from Wild Boars and Zombie Mushrooms
It is not recommended to do this quest until you can get your hands on a pair of Weighted Earrings, as they are not sold in shops, and they, like all equipment, are a very rare drop.