Learn Hardcore Spanish (Spain): Ese juego me divierte mucho.

The subject is ese juego. It is the thing doing the action of entertaining. Me is an object pronoun referring to the person who is entertained. In a more literal mapping to English: Ese juego = That game, me = me, divierte = entertains, mucho = a lot.

Can I also say Me divierte mucho ese juego? Does it change the meaning?

Yes, Me divierte mucho ese juego is also correct. The basic meaning is the same. Starting with ese juego sounds a bit more neutral or descriptive; starting with me slightly emphasizes the effect on you, as in That game really entertains me. In normal conversation both orders are common and natural.

Is me a direct or an indirect object here? Is divertir used like gustar?

With divertir, the person who is entertained is normally the direct object, so me functions as a direct object pronoun here. You could rewrite it (only for analysis) as Ese juego divierte a mí, where a mí is a direct object referring to a person. This is different from gustar, where the person is an indirect object: Ese juego me gusta literally means That game pleases me, and ese juego is the subject. So divertir is a regular transitive verb (like entretener), not a gustar‑type verb.