I hope I remember this well, it's been at least 10 years
Right-click on the field; you will see a planning layout there with month, year and little icons showing crops and grey icons showing silos. When the time reaches the date where such a silo icon is situated, the field will be harvested. You can use the buttons next to the timeline to add icons to the timeline (for example if you want to harvest next week already).
However when you just plant a field, the timeline will already include dates for harvest. So you better have a good reason to harvest earlier (for example the crop has already matured quite a bit and the prices will likely drop soon; or the crop has been eaten by locusts so you want to plant something new instead).
Make sure that you have seeds, fertilizer, pesticide, fungicide, herbicide available, otherwise it's more expensive.
Also, there is a help file in the game (see the right side of the menu, you should find help there).