Your farm start out as Beginner level. The next level is Amateur, then Experienced and so forth. Gaining a farm level will enable your farm to purchase more land, to gain more transactions on the market, to have more and better employees and to purchase new tools. To gain a farm level, you will have to obtain some accomplishments and fame. Fame could be gained in different manners. You can either earn fame by completing missions, winning challenges or even by earning accomplishments. When the requirements are met you should be able to level up your farm in your Accomplishment interface.
Accomplishments are separated into 3 categories, Livestock, Crops and Workshops. In order to level up your farm you will have to earn a certain set of accomplishments of your current farm level. For example, the Beginner Chicken Egg Accomplishment, in order to earn it, you will have to collect the indicated number of Chicken eggs of the specified quality. When you have collected the specified quantity, click the the accomplishment to earn it. You will gain a bit of fame too. Upon reaching the Amateur farm level you will be able to accomplish the Amateur Chicken Egg Accomplishment. Note that you will not score points on higher level Accomplishments if your Farm Level is inadequate.