"My father is teaching me to cook."
Translation:Mi padre me enseña a cocinar.
The "me" has to be before the conjugated verb. The only times you can have it attached to the verb is when you use the infinitive or a gerund. So you could have: Mi padre está enseñandome a cocinar. Or you could use future tense with: Mi padre va a enseñarme a cocinar.
In both of these cases, you could also just put the "me" before the conjugated verb. Mi padre me está enseñando a cocinar. Mi padre me va a enseñar a cocinar.
You cannot have an infinitive verb without first having a conjugated verb. So while not exactly the same meaning, you could say: Mi padre va a enseñarme a cocinar. However this sentence implies future tense, though, versus right now.
In another exercise (and similar ones), why is the boy "is learning to read" translated using the present progressive ("estar -ando") conjugation, and, in this exercise (and other similar ones), we just use the standard present tense conjugation? I'm having trouble figuring out when to use each one?
In Spanish, you usually can use either one interchangeably. I think there is a slight difference in meaning having to do with the length of the action (just this moment vs continuing and ongoing), but I don't recall which is which.
Many Spanish verbs need prepositions where there is no preposition in English, and vice versa. (Or the preposition isn't a direct translation of the English one.) You may find this page helpful. http://laspreposiciones.com/verbs-and-prepositions.html
It has to do with whether the object or direct object of the sentence is a person or an object.
La cocina is the noun, meaning kitchen. Cocina is the conjugated form of "to cook" for el, ella, and usted. Cocinar is the infinitive form of the verb (to cook).