You can choose:
Ich mache nächstes Jahr eine Ausbildung.
I'm doing an apprenticeship next year.
Ich mache im nächsten Jahr eine Ausbildung.
I'm doing an apprenticeship in the next year.
Let's start with case 2, because it's easier:
The part »im nächsten Jahr« is a temporal prepositional group. As a prepositional group, it begins with a preposition and the rest is an inner object, that is in a grammatical case, that is defined by the preposition. The word »im« is a contraction of the preposition »in« and the definite dative masculine or neuter article »dem«. The preposition »in« could go with accusative too, but only with movements (even temporal movements: »Irenes Rückkehr fällt schon in das nächste Jahr.«), but here we don't have a movement but a (temporal) place, and therefore we need dative case, and the article »dem« really indicates this case.
But in sentence 1 we don't have a preposition, and that makes things a little bit complicated:
The part »nächstes Jahr« is a temporal adverbial phrase, and it is in accusative case which you can read from the adjective which would have different endings in genitive and dative case. (It can't be in nominative case, because it's not the subject and its also not a »Gleichsetzungsnominativ« which would need to be connected to the subject via a copula verb like »ist«.)
So the question is: Why is it in accusative case? The case is not ruled by the verb and there is also no preposition that would rule the case.
The answer is: Nothing rules the case of this phrase. We use it in accusative case only, because we are used to use it in that case. It is a fixed phrase with a fixed grammatical case.