Tap on the icon which is a little trowel/shovel under a flower - the icon will show a number on it to indicate at a quick glance how many seeds you have to plant. This will open the seedling menu where you can then drag items out to one of the cleared squares in your garden. Plants you have already discovered will show a little picture of the plant type but you can also have mystery seeds which you won't know what they are until you fully nurture them!
Each plant has to be nurtured a certain number of times by a Gardener Bee before it grows and can be harvested. This is indicated tapping on seedling needing nurtured.
Plants like vegetables will be harvested once and then you will need to plant another one whereas flowers will remain. Harvesting vegetables will earn you more vegetable seeds for planting though. Vegetables are used for recipes and the Community Barter Board. Flowers are how you generate nectar for Expeditions!
You can move existing plants around in your Garden using the Seedling menu as well, simply open the menu and then tap a plant in the Garden to activate it and you will then be able to move it to another square.