1. First of all, you will need a farm. To buy one you have to bid on it just like you bid on Houses.
You have to enter house bidding menu and then switch to farms.

2. You get clay watering can from lvl 275 reward. If you misplaced it, you can always buy another one from the Seed Merchant.

3. To get to Seed Merchant you need to talk to Horse. He's located at -1 floor in DP building.

4. When you have your farm, watering can and seeds ready, you can start your farming frenzy.
To plant seed simply click on fence of your farm and select what you need to plant, or drop a single seed
on empty farm tile to plant it. Before planting you might want to visit Daily farming npc .

5. Then you have to water your plants and wait. To water plants you can either use watering can on them, or simply right-click them (with watering can in backpack).
Base growth time is 8 hours. This can be reduced by using Dragon Watering Can and futrther increasing your farming level.

6. When the time comes to harvest you will need a scythe (it can be also used directly on plants before they're fully grown to remove them).
Either use scythe on grown plant or simply right-click it while scythe is in your backpack.

7. Every harvest will cost you 10 energy that can be lowered by certain Trinkets, Backpacks and Companions.
These bonuses are additive and energy reduction items will definitely help you save alot of money.

To progress faster you can do daily tasks. Each day you will be asked to harvest specific plants. They will yield 200% of regular experience.