Learn French: La chèvre de la ferme est blanche.

In English, you can put two nouns together and use one like an adjective to describe the other (like farm goat). French rarely does this. Instead, you usually connect them with de (of). So, the farm goat literally becomes the goat of the farm (la chèvre de la ferme).

Why is the word for white spelled blanche here instead of blanc?

Adjectives in French must agree with the gender of the noun they describe. Because la chèvre (the goat) is feminine, you need the feminine form of blanc. This adjective is slightly irregular, adding an h before the e to make blanche.

Could I just say la chèvre de ferme without the second la?

It depends on the exact meaning you want! If you drop the article and say de ferme, it acts as a general category (a farm-type goat, as opposed to a mountain goat). By using de la ferme, you are being specific: it is a goat that belongs to a specific farm (the goat of the farm).