In order to plant and harvest your vegetables, first thing you neet to do is go to the plan planning table in the right upper corner of your house.

After you decided, where to build your field, select the field tile and draw the fields on the plan (the orientation only makes a visually difference). The price per field-tile is 50 gold.

You should also think about the way you want to water your plant. There are two options to water your plants. Read more to that at chapter 'Water supply for your plants'.

When you are done, save your changes and leave your table. Then leave the house and go to the field.

When you are on an empty field, you can see to currently selected plant in your quicktoolbar which you can plant usind the shown key. next to it you can see the symbol to change the currently selected plant. press the key shown under the changeing-symbol (In my case, playing with mouse und keyboard, it is the right mouse button)
Then plant the seeds using the shown key (In my case it is the left mouse button).

Next thing is waiting. wait until the plants are ready to harvest (and make sure they are always watered in the meantime).

When they are ready to harvest, they get the usual little collect icon above them. When this happens, go to the field and harvest the finished plant.
After you harvested it, the circle begins again, plant a plant, wait and water it and then harvest it.

Whenever your plants show this icon and the field is more light than dark, your plants need some of that fresh tasty water. There are three methods to water your plants when they need it ... at least theoretically.

The first (and dumbest thing is to wait for rain). Seriously why would you wait for the rain to water your plants? this is so inefficient. But if it happens to rain you don't have to worry about watering your plants, so see it more as a sometimes happening bonus.

The second way to water your plants is cheap, but it takes some time.
Refill your waterbucket and go to every field that needs water and water it manually. If you only have a few fields that might sound easy enough, but when the number of your field reaches 30 or even more this becomes a real pain in the butt.
If you decide to use this methode, you should keep an eye to a close enough water source so you dont have to run 10 miles each time you need to refill your bucket. So if you dont't have your field next to a river make sure to build a well next to it. You can build a well via the building table.

The most expensive but easiest way is to build sprinklers. They cost 1500 gold but they constantly water the fields around them. Each sprinkler can water a 3x3 tiles field just put it in the middle of it and it will instantly start it's duty. (You can build them on top of a field so you really have nine tiles done with one sprinkler.)
The downside of the sprinklers, besides their high acquisition cost is that they need power to work (but no water ...) so you have to create power with a fodder more often than without them. But I think it still isn't that time consuming as the water bucket method is