"My son started studying Spanish three years ago."
Translation:Mi hijo comenzó a estudiar español hace tres años.
Some verbs need a preposition before a noun or another verb. Here are a few: https://www.fluentu.com/blog/spanish/spanish-verbs-with-prepositions/
Two points: First, don't forget the accent (empezó) to make the past tense third person, or it can be confused with a first person present tense verb.
Second, the verb empezar must be followed by a when it refers to an infinitive verb. So your sentence should be:
Mi hijo empezó a estudiar español hace tres años.
No. This is one of those situations where we use the present participle in English ("studying") but you have to use the infinitive (estudiar) in Spanish.
This is going to be the case in almost all situations except for creating the progressive tenses (so "He is studying now" is Él está estudiando ahora). But for "He likes studying" or "He started studying" or "Studying is difficult" or "Studying is important" all take the infinitive instead.